NONLINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS OF MULTILAYER SHELL STRUCTURES BY XIANGGUANG TAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2002 To my parents. ACKNOWLEDGMENTS I wish to express my sincere gratitude to my advisor, Prof. Loc Vu-Quoc, for his pa- tience, guidance, support and friendship throughout my Ph.D. education at the University of Florida. I have greatly benefited from his stimulating approach to research and his re- lentless pursuit of perfection in organization and documentation. Many thanks are extended to him for his invaluable help in preparing this LINUX/LaTeX document. I also wish to acknowledge the members of my examining committees, Professors Martin A. Eisenberg, Raphael T. Haftka, Marc Hoit, Andrew J. Kurdila, and W. Gregory Sawyer for their careful examination of the dissertation, and their invaluable comments and insights, which made a deep impact on my research. I also benefited greatly from their graduate courses and from their help in many other aspects. I am indebted to several colleagues and mentors for their help in my present work: in particular, Hui Deng for the use of finite element code, FEAP and many insightful dis- cussions on the geometrically-exact shell theory; Fuller L. Brian for the installation of the LLNL package; Paul Dionne, Andrzej Przekwas, Marek Turowski, and H.Q. Yang at the CFDRC for discussion of the model reduction technique; Prof. Chen-Chi Hsu to work for him as his teaching assistant; and to my friends, Joakim Andersson, Jonas Bjornstrom, Mat- tias Horling, Stefan Jansson, Kil-Soo Mok, Mattias Quas, Simon. Sjogren, Xiang Zhang, and Yuhu Zhai, and many others, who have made my stay at Gainesville one of the most memorable periods of my life. Last, but certainly not least, my heartfelt thanks go to my parents for their love, and encouragement through my life. I am indebted to my girlfriend, Veronica Leung. Without her love and care, I could not have accomplished so much. This research is supported by a grant from the National Science Foundation, and also by the CFDRC. This support is gratefully appreciated. iii TABLE OF CONTENTS page ACKNOWLEDGMENTS iii ABSTRACT viii CHAPTER 1 OVERVIEW 1 1 . 1 Objectives and Motivation 1 1.1.1 Formulation and Kinematics 2 1.1.2 Computational Aspects 4 1.2 Chapter Overview 5 2 GEOMETRICALLY-EXACT SANDWICH SHELL FORMULATION 7 2.1 Introduction 7 2.2 Virtual Powers 8 2.2.1 Basic Kinematic Assumptions and Configurations 9 2.2.2 Virtual Powers 11 2.2.2.1 Power of contact forces/couples and conjugate strain mea- sures 11 2.2.2.2 Power of assigned forces/couples 13 2.2.3 Constitutive Relations 13 2.3 Weak Form and Linearization 14 2.3.1 Admissible Variations, Tangent Spaces 14 2.3.2 Weak Form of Equations of Equilibrium 15 2.3.3 Contact Weak Form 16 2.3.4 Assigned Weak Form 16 2.3.5 Linearization of Contact Weak Form 17 2.3.5.1 Update of inextensible directors 17 2.3.5.2 Perturbed configuration 18 2.3.5.3 Linearized strain measures 20 2.3.5.4 Linearized contact weak form 22 2.3.6 Matrix-Operator Format of Contact Weak Form 22 2.3.6.1 Material tangent operator 31 2.3.6.2 Geometric tangent operator 34 2.4 Numerical Examples for Statics of Sandwich Shells 46 2.4.1 Roll-down Maneuver of a Sandwich Plate 48 iv 2.4.2 Sandwich Plate with Ply Drop-offs 49 2.4.2.1 Sandwich plate with ply drop-off 49 2.4.2.2 Two-layer plate with ply drop-off: aspect ratio A = 5 : 1 : (1,0.5) 53 2.4.2.3 Two-layer plate with ply drop-off: aspect ratio A = 20 : 1 : (1,0.5) 53 2.4.2.4 Two-layer plate with ply drop-off: aspect ratio A = 20 : 10 : (1,0.5) 61 3 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER COMPOSITES : STATICS 68 3.1 Introduction 68 3.2 Kinematic Assumption and FHW Variational Formulation 74 3.2.1 Kinematics of Solid-Shell in Curvilinear Coordinates 74 3.2.2 Variational Formulation of EAS Method 78 3.3 Finite-Element Discretization 83 3.3.1 The Weak Form of Modified Two-Field FHW Functional 83 3.3.2 Spatial Discretization 84 3.3.3 Linearization of the Discrete Weak Form . . . 85 3.3.4 Material Law in Convected Basis 89 3.3.5 The ANS Method 93 3.3.5.1 Transverse shear strains 93 3.3.5.2 Transverse normal strain 94 3.4 Interpolation of the Enhanced Strains 94 3.4.1 The Regular Enhanced Strains Treatment 95 3.4.2 Proposed Efficient Enhancing Strains 100 3.4.3 Equivalence Between EAS Element and Incompatible Mode Element 102 3.4.3.1 Tensor form of enhancing strains 103 3.4.3.2 Equivalence of condensed stiffness matrices 107 3.5 Numerical Examples 109 3.5. 1 Patch Tests and Optimal Number of Parameters 110 3.5.1.1 Membrane patch test Ill 3.5.1.2 Out-of-plane bending patch test Ill 3.5.2 Cantilever Plate 113 3.5.2.1 Cantilever beam: in-plane bending 114 3.5.2.2 Cantilever plate: out-of-plane bending 115 3.5.3 In-plane Bending Problem with Nearly Incompressibility 119 3.5.4 Snap-through of a Shallow, Cylindrical Roof under a Point Load . . 121 3.5.5 Pinched Hemispherical Shell 122 3.5.6 Multilayer Composite Plate 125 3.5.6.1 Two-layer composite plate: linear solution 125 3.5.6.2 Multilayer composite plate with ply drop-offs 126 3.5.7 Multilayer Composite Hyperbolical Shell with Ply Drop-offs .... 129 v 4 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER COMPOSITES : DYNAMICS 132 4.1 Introduction 132 4.2 Dynamics of Solid Shells by an EM Conserving Algorithm 133 4.2.1 Time Discretization on Dynamic Weak Form 134 4.2.2 Linearization of Dynamic Weak Form 136 4.3 Enhanced-Assumed-Strain Method Based on Deformation Gradient . . . 143 4.3.1 Weak Form 143 4.3.2 Finite Element Discretization and Linearization 147 4.3.3 Assumed Natural Strain (ANS) Treatment 150 4.3.4 Simplified Formulation 151 4.4 Numerical Examples 153 4.4. 1 Double Cantilever Elastic Beam under Point Load 154 4.4.2 Pinched Cylindrical Multilayer Shell 157 4.4.3 Free-Flying Single-Layer Plate 159 4.4.4 Free-Flying Multilayer Plate with Ply Drop-offs 161 5 EFFICIENT AND ACCURATE MULTILAYER SOLID-SHELL ELEMENT: NON- LINEAR MATERIALS AT FINITE STRAIN 175 5.1 Introduction 175 5.2 Nonlinear Material Law 180 5.2.1 The Mooney-Rivlin Material Models 180 5.2.2 The Hyperelastoplastic Model 182 5.2.2.1 Multiplicative decomposition of the deformation gradient F 183 5.2.2.2 Spectral form based on the right Cauchy-Green tensor C . 185 5.3 Explicit Time Integration Method for Solid-Shell Elements 193 5.4 Numerical Examples 196 5.4.1 Large Deformation of Rubber Shells 197 5.4. 1 . 1 Stretch of a rubber sheet with a hole 1 98 5.4.1.2 The snap-through of a conic shell 198 5.4.1.3 Large motion of the pinched cylindrical shell 200 5.4.1.4 Rubber hemispherical shell 203 5.4.2 Large Deformation of Elastoplastic Shells 204 5.4.2.1 Bending of a cantilever beam 206 5.4.2.2 Elastoplastic response of a channel beam 208 5.4.2.3 Pinched hemisphere 211 5.4.2.4 Elastoplastic response of a simply supported plate 213 5.4.2.5 Elastoplastic response of a pinched cylinder 215 5.4.2.6 Free-flying multilayer plate with ply drop-offs 218 5.4.2.7 The impact of a boxbeam 221 5.4.2.8 Pipe whip 223 6 SOLID SHELL ELEMENT FOR ACTIVE PIEZOELECTRIC SHELL STRUC- vi TURES AND ITS APPLICATIONS 228 6.1 Introduction 228 6.2 The Solid-Shell Formulation 231 6.2.1 The Kinematics of Piezoelectric Solid-Shell Formulation 231 6.2.2 Piezoelectric Solid-Shell Element 234 6.2.2.1 Functional and finite element formulation 234 6.2.2.2 Linear piezoelectric material law in convected coordinate . 239 6.2.3 Composite Solid-Shell Element 242 6.3 Simulation Control Design 244 6.3.1 Finite Element System Equation of Piezoelectric Structure 244 6.3.2 Reduced-Order Model of Piezoelectric Finite Element System . . . 246 6.3.3 Controller Design 249 6.4 Numerical Examples 252 6.4.1 Cantilever Plate: Out-of-Plane Bending 253 6.4.2 Multilayer Composite Hyperbolical Shell . 255 6.4.3 Piezoelectric Bimorph Beam 256 6.4.4 Cantilever Plate with PZT Actuators 259 6.4.5 Cantilever Plate with PZT Actuator and Sensor 263 7 CLOSURE 268 7.1 Conclusion 268 7.2 Directions for Future Research 270 APPENDED A SOLID-SHELL FORMULATION 272 A. 1 Finite Element Approximation of Solid-Shell Element 272 A.2 Solution Procedure of Nonlinear Equations 279 A.3 Explicit Integration Algorithm with EAS Method 280 A.4 Return Mapping Algorithm for J 2 Flow Theory with Isotropic Hardening . 280 A.5 Elastoplastic Moduli S-J 281 A. 6 Algorithmic Moduli for Return Mapping 282 B PIEZOELECTRIC SOLID-SHELL FORMULATION 285 B. l Model Reduction Algorithm 285 B.2 Solving Procedure on Control Design 285 REFERENCES 287 BIOGRAPHICAL SKETCH 302 vii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NONLINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS OF MULTILAYER SHELL STRUCTURES By Xiangguang Tan August, 2002 Chairman: Loc Vu-Quoc Major Department: Aerospace Engineering, Mechanics, and Engineering Science Firstly, the geometrically-exact sandwich shell formulation is developed to analyze sandwich shells undergoing large deformation. Finite rotation of the director in each layer is allowed, with shear deformation independently accounted for in each layer. The thick- ness and the length of each layer can be arbitrary, thus allowing the modeling of multilayer structures having ply drop-offs. The weak form of governing equations is constructed, and the linearization and inextensible directors update are derived. Numerical examples on elastic sandwich plates are presented to illustrate salient features of the formulation. Furthermore, we present a low-order solid-shell element formulation — having only displacement degrees of freedom (dofs) (i.e., without rotational dofs)— that has an opti- mal number of parameters to pass the plate patch tests (both membrane and out-of-plane bending), thus allowing for efficient and accurate analyses of large deformable multilayer shell structures. The formulation is based on the mixed Fraeijs de Veubeke-Hu-Washizu (FHW) variational principle leading to a novel enhancing assumed strain (EAS) tensor, with improved in-plane and out-of-plane bending behaviors (Poisson thickness locking). Shear locking and curvature thickness locking are treated using the Assumed Natural Strain (ANS) method. We provide an optimal combination of the ANS method and the minimal number of EAS parameters to pass the out-of-plane bending patch test and treat the locking viii associated with (nearly) incompressible materials. The energy-momentum (EM) conserv- ing algorithm for the current element is presented. Two nonlinear 3-D material models are applied directly without requiring the enforcement of the plane-stress assumption. More- over, we present a low-order accurate piezoelectric solid-shell element formulation for piezoelectric sensors and actuators used in active shell structures. Numerical examples in- volving static analyses and implicit/explicit dynamic analyses of multilayer shell structures having a large range of element aspect ratios for both material and geometric nonlinearities are presented. Numerical examples involving static analyses and active vibration control of piezoelectric shell structures are also presented. The developed element formulations are accurate and efficient in modeling and analyzing general nonlinear multilayer composite shell structures. CHAPTER 1 OVERVIEW Shells and shell structures are thin-walled, generally curved bodies in a three-dimens- ional space. Their load-bearing behavior is dominated by stretching and bending. Shell structures with different layers in the thickness direction are generally addressed as multi- layer shells. For a comprehensive and valuable history and review of linear and nonlinear shell theories, see Timoshenko and Woinowsky-Krieger [1959], Naghdi [1972] and Basar and Kratzig [2000]. Below we describe the objectives and motivation for the current re- search on multilayer shells. Some of the motivating factors behind the present work and literature review are delineated in the following chapters. 1.1. Objectives and Motivation Multilayer shell structures have widespread applications in engineering. Laminated composite structures, initially developed for use in the aerospace industry, have played an increasingly important role in robotics and machine systems that require high operat- ing speed. The low weight and high stiffness offered by laminated composite structures help reduce power consumption, increase the ratio of payload/self-weight, and improve the accuracy of motion characteristics and reduce the level of acoustic emission of these systems. It is shown from computer simulations with experimental corroboration that the low weight/stiffness ratio of laminated composites is essential for obtaining high perfor- mance in slider-crank and four-bar linkage systems (Sung, Thompson, Crowley and Cuccio [1986], Thompson and Sung [1986]). More recently, considerable attention has been given to a class of active structures with embedded piezoelectric layers as sensors and actuators (Evseichik [1989], Tzou [1989], Saravanos, Heyliger and Hopkins [1997]) or interfero- metric optical fiber sensors (Sirkis [1993]) for monitoring the strain level and for vibration control. Large overall motion of multilayer structures can be found in robot arms or space 1 2 structures with embedded sensors/actuators. Another example of multilayer structures can be found in the damping of structural vibration by using viscoelastic constrained layers (Al- berts [1993], Dubbelday [1993], Rao [1993]) (Figure 1.1). The use of sandwich plates to absorb energy in crashes (car, train, airplane) was investigated by Goldsmith and Sackman [1991]. Figure 1.1. Multilayer shells with patches of constrained viscoelastic materials or of piezo- electric materials. The design and analysis of multilayer shell structure is a major challenge that in- volves the proper modeling of composite materials with highly anisotropic properties, com- plex geometric configuration, and strongly nonlinear material behavior. For example, only a few studies so far have been performed on large deformation analysis of 3-D nonlin- ear composite laminates. There have been no analytical studies involving 3-D analysis of multilayer shells with nonlinear material behavior and large deformation. 1.1.1. Formulation and Kinematics For nonlinear analysis of multilayer shell structures, we developed two different finite element formulations: the geometrically-exact 1 multilayer shell formulation and multilayer solid-shell formulation. In the geometrically-exact multilayer shell formulation, the 3-D analysis is reduced to a set of 2-D stress-resultant equations based on the kinematic assumptions. This model accommodates large deformation and large overall motion. The layer directors at a point in 1 The term "geometrically-exact" reflects the fact that no additional kinematic assumptions are made beyond the one-director assumption. In particular, approximations of the type sinO es 9 - 3 /6 are entirely avoided. 3 the reference surface are connected to each other by universal joints, as in a chain of rigid links. The thickness and length of each layer can be arbitrary, thus making it suitable to model shell structures with ply drop-offs. The equations of motion of the multilayer shell are derived based on the principle of virtual power, and expressed in terms of weighted resultant forces and couples. The overall deformation of a sandwich shell can be described by the deformation of a reference layer (which can be any layer; not necessarily the middle layer). The unknown kinematic quantities are therefore the three displacement components of the centroidal surface of the reference layer and two rotational components for each layer director. No restriction is imposed on the magnitude of the displacement field, whose continuity across the layer interfaces is exactly enforced. Finite rotations of the directors in each layer are allowed, with shear deformation independently accounted for in each layer. We have implemented the geometrically-exact sandwich shell element to illustrate the ver- satility of formulation in the large deformable multilayer shell analysis involving linear elastic material and small strain. Due to the kinematic assumptions, the present formula- tion is more accurate than the equivalent single-layer shell models in the interlaminar stress analysis, especially for thick and moderate thick shells. In the solid-shell formulation, on the other hand, the shell kinematic descriptions used are the displacement of the top and bottom surface of the shell. All kinematic quan- tities such as displacements and the corresponding strains can be finite. For multilayer shells, one solid-shell element in the thickness direction can be used for either one material layer or several layers. In contrast to the shell formulation based on the degenerated shell concept and the classical shell theory, the present element can incorporate the complex 3-D material models without enforcing the zero transverse normal stress condition, can avoid complex update algorithms for finite rotations, and can account for the transverse normal stress. Based on the mixed Fraeijs de Veubeke-Hu-Washizu (FHW) variational principle, the present low-order solid-shell element is designed to pass the plate patch tests and to remedy volumetric locking, therefore allowing efficient and accurate nonlinear analyses of multilayer shell structures. Moreover, the kinematic description provides a natural way 4 to connect solid-shell elements to regular solid elements without the need for transition elements; such feature can also benefit the detailed modeling of shells with patches of piezoelectric or viscoelastic materials. For the interlaminar stress analysis, with the refine- ment through the thickness, the solid-shell element model can determine the localized 3-D stress field (e.g., delamination, free-edge effect) accurately. 1 . 1 .2. Computational Aspects Several aspects can directly contribute to the success and generality of numerical simulations: 1) element formulations; 2) time-integration schemes; and 3) equation solu- tion strategies. The geometrically-exact sandwich shell formulation uses the resultant form to avoid numerical integration in the thickness direction for elastic materials. The solid- shell formulation uses the numerical integration for general nonlinear constitutive models. All kinematic quantities such as displacements can be finite, and the update procedure is proceeded in an exact manner, without approximations. Engineering applications mandate the use of relatively coarse meshes for complex geometries. The development of convergent elements, which are free of spurious numer- ical locking, are variationally consistent, achieve good accuracy with coarse meshes, and satisfy stability and completeness requirements, is essential. Flexural super-convergence in membrane deformation is also important for applications involving in-plane bending. Moreover, the use of low-order interpolations is extremely desirable for their simplicity, efficiency and amenability to contact implementations. To this end, we use the methods of enhanced assumed strains (EAS) and assumed natural strains (ANS) judiciously to con- struct low-order elements possessing the above features for the analysis of multilayer shells. In this work, we have implemented a number of dynamic time-stepping implicit/explicit algorithms in the context of the present formulations for transient integration of the result- ing semi-discrete finite element equations. The time step-size for the implicit integration can be much larger than that for the explicit integration. The explicit method, on the other hand, needs much less computational effort at each time step since the matrix factoriza- tion is not needed. For elastodynamics, the introducing of numerical damping is essential 5 to increase the numerical stability of implicit integration methods, even for the energy- momentum conserving algorithm. The solution of discrete equations for problems involving large deformation and long-term simulations can be accomplished with the Newton-Raphson scheme. The numer- ical efficiency of this approach is a byproduct of the asymptotically quadratic convergence of its iterations. To maintain this rate of convergence, the exact linearization of discrete equations is explicitly obtained and implemented in the present work. For nonlinear mate- rials, the consistent tangent moduli are crucial to be derived. For the quasi-static analysis of unstable systems, arc-length method is used to find stability points and trace post-buckling paths. Based on the above solution strategies, a large time or load increment is allowed to use, while a good balance of accuracy and efficiency is maintained. 1.2. Chapter Overview This dissertation is divided into six chapters. Two finite element models for multi- layer shell structures, the geometrically-exact sandwich shell element and the solid-shell element, are formulated and implemented. Chapter 2 presents the static analysis of the geometrically-exact sandwich shell el- ement formulation. The kinematic description and equilibrium equations of the sandwich shell model are presented in Section 2.2. The corresponding weak form and linearization are given in Section 2.3. Numerical examples for statics of sandwich shells are shown in Section 2.4. This chapter has been published by Vu-Quoc, Deng and Tan [2000]. Readers refer to Vu-Quoc, Deng and Tan [2001] for the corresponding dynamic analysis. In Chapter 3, we carry out the static analysis of the optimal solid-shell element for- mulation for multilayer composites. After a presentation of the kinematics assumption and the formulation based on the Fraeijs de Veubeke-Hu-Washizu (FHW) variational prin- ciple (Felippa [2000]) in Section 3.2, we discuss the finite-element discretization and its implementation in Section 3.3. A review of the EAS method together with our proposed modification is presented in Section 3.4. We present the numerical results in Section 3.5. This chapter will be published by Vu-Quoc and Tan [2002a]. 6 Chapter 4 addresses the dynamic analysis of the optimal solid-shell element formu- lation for multilayer composites. We devoted Section 4.2 to the dynamic aspect and the use of the energy-momentum algorithm for elastic materials. A variant of the EAS formulation based on the deformation gradient (instead of Green-Lagrange strains) for solid shells is the focus of Section 4.3. Numerical results are shown in Section 4.4. This chapter will be published by Vu-Quoc and Tan [20026] . In Chapter 5, we present static and dynamic analyses of the multilayer solid-shell element formulation for nonlinear materials at finite strain. Two nonlinear material models (i.e., Mooney-Rivlin material and hyperelastoplastic material), and their implementations are discussed in Section 5.2. The explicit integration method for solid-shell elements is addressed in Section 5.3. Numerical simulations, which illustrate the performance of the proposed element formulation, and exhibit both material and geometric nonlinearities in the large-scale implicit/explicit analyses, are given in Section 5.4. This chapter was submitted for the publication by Tan and Vu-Quoc [2002a]. Chapter 6 discusses the solid shell element for active piezoelectric shell structures and its applications. In Section 6.2, we introduce the kinematics and variational formu- lation of the piezoelectric solid-shell element, and then present the composite solid-shell element. The control design for structures with piezoelectric sensors and actuators is dis- cussed in Section 6.3. Numerical simulations that illustrate the performance of the pro- posed formulations, including comparisons with available experiment results and solutions obtained from shell elements and solid elements, are given in Section 6.4. This chapter was submitted for the publication by Tan and Vu-Quoc [20026]. Chapter 7 gives the closure of our work. Conclusions are drawn in Section 7.1 and directions for future investigation are suggested in Section 7.2. CHAPTER 2 GEOMETRICALLY-EXACT SANDWICH SHELL FORMULATION 2.1. Introduction Sandwich structures have played an important role in several areas of engineering. Many background references were cited by Vu-Quoc and Ebcioglu [1995], and are not re- peated in the present follow-up work, except for particularly relevant ones. We refer to review papers such as Reddy [1989], Noor and Burton [1989], Noor [1990], Reddy and Robbins [1994], and the references therein for various aspects on formulations for multi- layer structures. The accuracy of layerwise theory, as compared to single-layer theory with a shear correction factor, was demonstrated amply by Reddy [1989], where a comparison of transverse shear stress with 3-D elasticity solution was provided (Reddy [1993]). We de- scribe here a continuation of the results reported by Vu-Quoc, Ebcioglu and Deng [1997], where the equations of motion for geometrically-exact sandwich shells are derived. Focus- ing on the static case in the present work, we develop a Galerkin projection of the resulting nonlinear governing equations of equilibrium. In the present formulation, each layer in a sandwich shell structure can have different thickness and side lengths. As such, the present formulation can be used to model an important class of multilayer structures with ply drop-offs. Another important application of the present formulation is the modeling of shell structures with patches of constrained viscoelastic materials and/or patches of piezoelectric materials. No restriction is imposed on the magnitude of the displacement field, whose continuity across the layer interfaces is exactly enforced. Finite rotations of the directors in each layer are allowed, with shear deformation independently accounted for in each layer. The layer directors at a point in the reference surface are connected to each other by universal joints, and form a chain of rigid links. The overall deformation of a sandwich shell can be described by the deformation of 7 8 a reference layer. The unknown kinematic quantities are therefore the three displacement components of the deformed reference surface, and the unit directors associated with the layers. The starting point for the development of the Galerkin projection of the governing equations of equilibrium is a nonlinear weak form based on the stress power of a sandwich shell, from which the expressions of fully nonlinear strain measures are obtained (Vu- Quoc et al. [1997]). A linearization of this nonlinear weak form is performed for use in the solution for the kinematic quantities via the Newton-Raphson method. Together with the update of the inextensible directors, the linearization leads to a symmetric tangent stiffness operator, which is composed of a geometric part and a material part. The consistency in the linearization leads to a quadratic rate of asymptotic convergence in the Newton- Raphson iterative solution. Linear finite element functions are chosen to form a basis for the Galerkin projection of the linearized equilibrium equations into a finite-dimensional subspace of trial solutions. The tangent stiffness matrix is symmetric, and is evaluated using selectively reduced integration in all layers to avoid shear locking. Several numerical examples, including the bending and torsion of a sandwich plate, are presented to illustrate the salient features of the present formulation. In particular, the important case of sandwich shells with ply drop-offs under large deformation is presented. Results are compared with those obtained using the commercial nonlinear finite element code ABAQUS [1995]. 2.2. Virtual Powers In this section, we summarize the kinematic description of the sandwich shell model developed in Vu-Quoc et al. [1997], and the equilibrium equations in weighted resultant form. 2 The component form of the stress power and of the constitutive relation are also given at the end of this section. 2 The word "weighted" here is used to indicate that the resultants are in "weighted" tensor form and not in true tensor form (the reader is referred to Vu-Quoc et al. [1997] for an explanation). 9 2.2.1. Basic Kinematic Assumptions and Configurations 1 Let A designate the material surface of the shell, and H := (J t£\H the total thickness of the sandwich shell (Figure 2.1), where (qH is the thickness of layer (£). Let <&o : A x H i— > B Q be the mapping from the material configuration to the initial (reference) configuration, where A x H = B is the material configuration of the sandwich shell. The material domain for layer (£) is denoted by ^B such that l W B :=Ax [t) H , B:= |J (e) B . (2.1) t=-\ 2 Figure 2.1. Sandwich shell: Profile and geometric quantities. Let $ : A x H i-> B t be the deformation map from the material configuration to the current deformed (spatial) configuration. We use the notation £ := { £ 3 } e B to denote the coordinates of a material point, where £- := { f 1 , ^ 2 } 6 .4 is referred to as the material surface coordinates, and £ 3 e H the material through-the-thickness coordinate. The deformation map for each of the three layers is written as follows »*(*,*):= (e)V(e,t) + (e - {t) Z) m t , for ^= -1,0,1, (2.2) 10 where ^<p : A h* R 3 is the deformation map of the centroidal surface of layer (£), and (£)t : A ' — * 5 2 the unit director (represent transverse fiber vector) associated with layer (£), (f)Z the distance from the centroidal surface of layer (£) to the centroidal surface of the reference layer (0) (for which (o)Z = 0). The centroidal surface of each layer (£), which does not necessarily correspond to the geometric center of the cross-section of layer (£) are defined as follows: Let ^)p and ^)p t be the mass density in the initial and the spatial configuration, B and B t , respectively. We select the centroidal surface (e)<P of layer (£) such that / (e + (-dz) i-Dio ( - l)Po de = J (e + (-dz) (-^jti-DPtde =o, / £ 3 (o)j (o)Pod^ 3 = I £ 3 ( o)Jt wPtd? =0, (2.3) (o) 7 * (0)W / (^ 3 - (i)^) (i)i (i)P ^^ 3 = / (e 3 - m Z) (Dftd^ 3 =0. (1)W Using the assumption that the layer directors behave like a chain of rigid links connected to each other by universal joints, the deformation maps of the centroidal surface of the outer layers (1) and (-1) can be related to the deformation map of the centroidal surface of the reference layer (0) as (i)V (€ a , t) := (0 )V> t) + {0 )h + ( o)t + , (-i)V := ( )<p t) - (0) h~ (0) t - ( _i)t , (2.4) where and (^/i - are the distances from the centroidal surface of layer (£) to its top surface and to its bottom surface, respectively. The deformation of the sandwich shell is therefore described by four vector-valued mappings collectively denoted by * := {(0)<P, (e)t, for £= -1,0, l} . (2.5) In terms of components, we have three components for (0 )V?, and two components for each inextensible director w t, thus leading to a total of nine components, which are the principal kinematic unknowns to be solved for. 1 1 The deformation gradient for layer (I) is {t)F := GRAD w $ GRAD (£)$ . (2.6) The Jacobian determinants of the mapping (/)4> , and are given below {e) j := det[GRAD (0 * o (£)] , (2.7) w j ( := det[GRAD w * (£,*)]. (2.8) 2.2.2. Virtual Powers Here we summarize the expressions for the power 7 C of contact forces/couples and for the power 7 a of assigned forces/couples. Together these powers play a crucial role in the derivation of the equations of motion (see Vu-Quoc et al. [1997] for more details). The balance of the power of contact forces/couples and the power of assigned forces/couples as expressed by T c = CP a leads to the equation of equilibrium for geometrically-exact sandwich shells (see Eq. (84) and Eq. (85) of Vu-Quoc et al. [1997]). 2.2.2.1. Power of contact forces/couples and conjugate strain measures The set of convected basis vectors on the spatial (current) centroidal surface are de- noted by | (f)<Zj j, where the underlined index i is to be expanded in the following sense {w a i} : ={w a i> W a 2> (l)°3 } := { i (<)¥>,2> (<)*}| { 3 =0 ' for£=-l,0,l. (2.9) The co- vectors j ^ai }, dual to the vectors j ^a* }, are defined by the standard orthogonal relation < {e) a j , W 0i >= S{ , (2.10) where 5j is the Kronecker delta. The convected basis in the initial configuration is given by specifying t = in the spatial-centroidal-surface convected basis ^a* to obtain {{t)A L } := [ {i) Ai, {e) A 2 , } := { (QOi , w a 2 , W a 3}| t=Q , for£= -1,0,1. (2.11) 12 The basis j (t)A- }, dual to the initial-centroidal-surface convected basis { (e)Aj V, is de- fined similar to Eq.(2. 10). The membrane strain ^e, the transverse shear strain , and the bending strain measure (^p , which (as we will see later) are conjugate to the effective resultant membrane stress (t)^ 01 , the transverse shear ^q a , and the resultant couple (i)mP a , respectively, can be defined as follows := (f)€o0 w a Q <g> {e) a , (2.12) (e)8 ■= (t)6 aii )a a , (2.13) (t)P (e)Pap (e)a a $ (<)o". (2.14) The components of the membrane strain (^e, the transverse shear ^)5 a , and the bending strain w p are given in the following relations, respectively, (3*0*1 = ^ ((0 a <*/3 - , (<)*a = {i)la - {tft% i (e)Pa0 = (<)Ka/J ~ , (2.15) where (<)«*o0 : = M^'to^i := (i)A a -( t) A , (2.16) are the components of the Riemannian metric tensors of layer (£), ^a a0 w a a ® ^a and (<)>l Q/ j (^)A Q ® {t)Ar, in the current configuration and in the initial configuration, respectively (Naghdi [1972], Marsden and Hughes [1983]). The shear strain measures, which are measures of how much the director {l) t and the director {e) t Q depart from the normal to the centroidal surfaces in the current configuration and in the initial configuration, respectively, are defined as {i)la = (t)a a ■ {e) t, ^7° = (/) A a • {e) t . (2.17) Finally, the current and the initial (nonsymmetric) director metric {e) K a(3 and {e) K° af3 for layer (£) are defined as («)«<*/? := (*)a a • w t , {e) K° ap := (e) A a • {i) t 0>(3 . (2.18) 13 Using the above definitions, the stress power T c of geometrically-exact sandwich shells can now be written (Eq. (199) of Vu-Quoc et al. [1997]) as dA, (2.19) where (i)n al3 , (t)rh al3 and ^q 01 are the components of the weighted effective membrane force, the weighted resultant couple, and the weighted effective shear force, defined in (169), (174), (175) of Vu-Quoc et al. [1997] respectively. See the footnote at the beginning of this section for the meaning of the word "weighted." 2.2.2.2. Power of assigned forces/couples Let n* denote the distributed assigned force on the centroidal surface of the refer- ence layer (0), and (^m* the distributed assigned moment on layer (£). On the boundary dA, we assume that the normal to the lateral surface of the shell domain in the material configuration is such that (f) I/ = (0)1/ = ( )^a E a . (2.20) The assigned force n* a and assigned couple ^)fn* a on the boundary dA are then defined such that n* = n* a {Q) u a , {e) fh* = {e) rh* a {Q) v a . (2.21) The power of the assigned forces and couples is written as follows (Vu-Quoc et al. [1997, (56)]) "Pa = f (n* -u + (t)m* ' (<)< ) dA + I [n* a -u {Q) u a + £ (£) m* Q • {t) t {t) u a ) d(dA) . (2.22) d A V «— 1 / 2.2.3. Constitutive Relations For layer {£), we employ the following constitutive relations between the above strain measures and the mentioned effective resultant forces/couples W n - / r2~W- W e 7<5> ( 2 - 23 ) 1 -(it)V) 14 ~0a {1)3 1 (?) E [t) Hi -Pa-yS . r9 9 ^ w to m = — f TzTW^ WPiS i 12.-24; 12 = (<)Jt {t)tl {t)A {1)0 G inH mA al3 11)83 , (2.25) where {i)H := + w h~ , {e) A aP := {e) A a • (^A^ , (2.26) are the thickness of layer (£) of the sandwich shell, and the Riemannian metric tensor in the initial configuration for layer (£) of the sandwich shell. In Eqs.(2.23) to (2.25), m& is the Young's modulus, [i)G the shear modulus, the Poisson's ratio, and ^k s the shear 5 correction coefficient, for all layer (I). With mK s = -, Eq.(2.25) is the same as that given 6 in Naghdi [1972, p.587]. The elastic constant ^3 , with its component form given as W S*** = {e) u W A*> {£) A* + i (l - qu) qA* + {ei A(» {e) A^) , (2.27) is a fourth-order elasticity tensor. 2.3. Weak Form and Linearization In this section, we construct the weak form of the equations of equilibrium obtained in Vu-Quoc et al. [1997], and linearize this nonlinear weak form, which plays an important role in finite element implementation. The shell layers are assumed to be inextensible in the thickness direction, and there is no drill degree of freedom (dof) for the directors (i.e., the directors are not rotating about themselves) considered. We have three translational dofs for displacement of the centroidal surface of the reference layer (0), and two rotational dofs for the directors of each layer. For a sandwich shell, the total number of dofs is nine (9). For the single-layer case, we refer the readers to Simo and Fox [1989] and Simo, Fox and Rifai [1990] for the details. 2.3.1. Admissible Variations, Tangent Spaces The admissible variations to the deformation map <3> = ((o)V) (0)*, (1)*}. are denoted by 5 $ := {5 {0) ip , 8 , 5 (0) t , 6 m t } . (2.28) 15 Let TS denote the tangent space formed by the Cartesian product of the tangent spaces T i S 2 (i.e., the tangent spaces to the sphere S 2 at w t £ S 2 , for £ = -1,0, 1). We write l (2.29) The space of admissible variations, denoted by B t (i.e., the tangent space to the current configuration B t ), at the current deformation <fr , is then defined as T$ B t := :^h->IR 3 x TS\6 {0) <p = 0ond<pA, 5 {i) t =0ond (<)t .4} , (2.30) where dip A and d ^ A represent the portions of the boundary dA where the essential boundary condition is imposed on <p and on (^)t , respectively. 2.3.2. Weak Form of Equations of Equilibrium The weak form of the equations of equilibrium for sandwich shells is readily provided by the principle of virtual power, expressed by the following balance of power (Vu-Quoc et al. [1997]) y c = Pa , (2.31) where J* c and 7 a are the power of the contact forces/couples and the power of the as- signed forces/couples for the sandwich shell, respectively. It suffices to replace the time rates in the expressions for the powers y c and 7 a by the admissible variations S $ to obtain the weak form, which can now be written as Find * , such that G? c (*,<5*) = <?<,(**) , (2.32) for all admissible variations 6 <J> , where G c ( $ , S $ ) is the weak form of the contact forces/couples (or contact weak form for short), and G a (8 $ ) is the weak form of the assigned forces/couples (or assigned weak form for short). 16 2.3.3. Contact Weak Form The contact weak form of the contact power 7 C in component form is as follows: 1 » M 1 r n « • 1 G c ( $ , 5 # ) = 51 / 5 <5 (f)«a/? + + 6 {1)1 a dA 1 /■ ~ 1 = H / [w na/3(5 W e a/? + {t)m al3 5 w p aP + (e^S^Sa^dA, (2.33) where (£)n Q/5 , (e)rn a0 and (^)<f* are the components of the weighted effective membrane force, the weighted resultant couple, and the weighted effective shear force, defined in Eqs. (169), (174), (175) of Vu-Quoc et al. [1997] respectively, whereas 5 ^)e a p = (1/2)5 {t)a a0 , 5 (e)Pap — 5 (£)« Q /3 , and 5 ^S a = 5 (t)1a are respectively the variations of the strain mea- sures conjugate to the above weighted resultant tensors (Eq.(2.15)). The component form of the contact weak form is used in the computational formulation due to the constitu- tive laws Eqs. (2.23) to (2.25) that relate the weighted resultant tensors to their respective conjugate strains. 2.3.4. Assigned Weak Form From the power 7 a of the assigned forces/couples in Eq.(2.22), we obtain at once the weak form of the assigned forces/couples G a {6& ) m J ^n* 'S (o)V> + W** ' 5 W* j dA f 1 dA L «— 1 d(dA) . (2.34) Remark 2. 1 . For the dimension of the assigned forces/couples n* and (i)fn* , we refer the readers to Vu-Quoc et al. [1997]. At the boundary dA of the sandwich shell, n* and (f)Tn* are decomposed as follows n* = n* a v a on d n A, w m* Q {e) u a on d m A , (2.35) (2.36) 17 where d n A and d (t)Tn A are the portions of the boundary where the assigned forces and the assigned couples are applied, respectively. I 2.3.5. Linearization of Contact Weak Form To construct the linearization of the contact weak form Eq.(2.33) at a given configu- ration $ in the direction of an incremental tangent field: A $ := (A ( o)¥> , A ( _D« , A (0) t , A m t ,)er # 5, (2.37) we consider a one-parameter family of perturbed configurations £ h-» * e = ( (0 )^ e , (_l)* e > (0)*e . (1)** ) , (2.38) such that &e L = * < =A*. (2.39) The tangent contact weak form will be shown to be composed of a material tangent stiffness operator and a geometric tangent stiffness operator. The linearization of the contact weak form plays a central role in the computational procedure based on the Newton-Raphson method. 2.3.5. 1 . Update of inextensible directors The following steps are used in the update of the inextensible layer directors. 1. First, we must account for the assumption that the layer directors have no drilling dofs in their increments. The removal of the drilling dof is realized by nullifying the com- ponent A W T 3 of the material incremental directors A (/)T along the basis vector E 3 , that is A w r =A {e) T a E a , (2.40) where Greek indices take values in {1, 2}. Let the superscript fc on a tensor quantity denote the fcth iteration in the Newton-Raphson procedure. The spatial incremental director A ( t )t k for layer (£) is related to its material counter part A ^)T k through the orthogonal tensor w A fc as follows A(^«* = (() A*'A W Z* . (2.41) 18 If we do not make the distinction between tensors and their matrices of components, and the quantities in Eq.(2.41) in terms of matrices of component, then Eq.(2.41) can be written as A [e) t k = {e) A k A {e) T k , (2.42) where the matrix (£)A fc 6 R 3x2 is formed by the first two columns of the matrix (^A fc E ra3x3 2. The spatial director is updated as follows {e) t k+1 :=exp . k sin 1 1 A^t k = cosH A w t* || w t k + " % " A {l) t k . (2.43) II A W* II 3. The incremental director rotation matrix is obtained from the exponential map A w A fc :=exp 50(3) [ (t) 0] = cos|| {t) || 1 + sin|| {t) 9 || e + [l - cos (|| (t) ||)] e ® e , (2.44) where 1 is the identity tensor, [J and e are skew-symmetric tensors with and e as their associated axial vectors respectively, {t) := w t* xA w t* , e := F ^- J . (2.45) II II 4. Update the rotation matrix of layer (£) w A fc+1 «A W A* W A* . (2.46) The above procedure is very important for the linearization of the weak form. 2.3.5.2. Perturbed configuration Let A<I> := |A(o)V, A(_j)t, A( )t, A^tj G T<j> <S be the incremental field in the tangent space at the current configuration $ . The perturbed configuration along the increments in the tangent space is defined as follows *e = { (0)<P e . (-!)*« ' (0)*e , } , (2.47) 19 with (0)< {e) t £ := exp (<)t eA {l) t , for I = -1, 0, 1 , (2.48) (2.49) where r . I a n sm ll £ Aie)t || ex P w t [ eA M*j :=cos ll £A W* 11(0* + || £A{e)t || > (2.50) is the exponential map from T i S 2 to S 2 (see Simo and Fox [1989] for details). We now verify that Eqs.(2.48) and (2.49) satisfy (2.39). First, it is obvious from (2.48) and (2.49) that (0)V>« £=0 = (ojV e=0 = {t) t for £=-1,0,1 (2.51) (2.52) Second, by taking the directional derivative of (2.48) and (2.49), we obtain = A (0) v> , d de (o)¥> £ d de (2.53) £=0 £ = -|| A {e) t ||sin(J| eA«)* ||j w t + || A (0 t ||cos [\\ eA (e) t = for/ = -1,0,1. Next, to linearize the strain measures, the following formulas are useful A w t A w t e=0 (2.54) de (o)V> e £=0 = A m t (2.55) e=0 With the above results, we can now proceed to the linearization of the strain measures followed by the linearization of the contact weak form. Remark 2.2. The directional derivative of 8 ^t E along the direction of the increment A (^)i can be expressed in terms of the variation 8 y)t and the director as follows. It 20 is clearer if one thinks of the symbol 5 in the variation 5 ^t e as a derivative with respect to some variable, whereas the perturbation parameter e is a different variable. Since || \\ = 1, it follows that mi • (e)t = 0, and we can define (qv := x such that i as w o> x w t , or 6 fat = <5 w x , (2.56) where ( f )U> and S ^0 are vectors that play the same role, one for the time derivative, while the other for the variation of (qt . For the perturbed director ^)t E , we have 6 (i)t £ = 8(t)0 x ( f )t e , (2.57) with the same rate 8 ^0 as for Hence d de 8 (t)t e 5(i)0 x £ = de = <5 W x A m * £ = -(5 w t-A (£) t) ( t) t, far/ --1,0,1. (2.58) Since which is a result of (2.56) 2 and || w t ||= 1, and since • A w t =* 0. (2.59) 2.3.5.3. Linearized strain measures Let ( (e)a E a i (<)7a/3 > ) t> e the strain measures corresponding to the perturbed configuration (2.47), as defined in (2.16)— (2.17), and let the incremental strain measures be defined as (A (fidof, , A W 7 a/J , A (QKan) := — ( ( t )a e a0 , ( £)7 Q /? . ) £ = for -1,0,1. (2.60) We obtain the following expressions A (e\CL (2.61) 21 A W 7a := (A (<) t • (e) <p <a + { t)t 'A {e) <p a ) , (2.62) &(QKa0 ■= •(/)*/» + (t)*P, a ' A w t l/? ) , for £=-1,0,1. (2.63) For £ = 1 and I — — 1 (i.e., the top layer and the bottom layer), we want to express the incremental strains A^a a 0, A^)j a , A^)K al3 in terms of the deformation map <fr := | ( o)¥?, (_i)t, (o)*, (i)*} and its increment A <E» :={A(o)¥?, A(_i)t, A (0) t, A (1) t}. This objective can be achieved by employing the constraints (2.4) in (2.63), and then we obtain A (1) a Q/3 = (A (0) v?, q ■ ( o)fP,0 + (0)¥>, a "&(o)<P,p) + (0)h + (A (0 )VJ, a * ( O )t,0 + A( )<,/3 * ( )V,a + A(0)V,/3 ' (0)*a + (O)V 5 ,/? ^(o)i«) + (A ( )V,a * + A {1 )tp « (0 )V?, a + A (0 )V> i/3 ' (l)t a + (0)^ * A (l)£,a ) + (0)^ + (A(0)*,a • + (0)*,a *A(i)t^ + A(i)t >a • ( O )t,0 + (l)*a *A( )t,/?) + ((0)^ + ) (A(0)t, a • (O)t,0 + (0)*q "A( )t /?) + ((!)/»-) (A A(l)7a = A (0 )¥>, Q • ( )t + ( )V,a •A(O)* + (0)/i + (A(0)t a * (1)* + (o)*,a*A(i)*) + ( )/i _ (A ( i)t a »(i)t + ( i)t Q «A ( i)t) , (2.65) A(i)« a /J = A (0 )V,a • (1)*^ + (0)<P,a *A(l)*,/3 + (0)^ + (A (0)t a * (1)*,/? + (o)*,a "A(i)*/j) + (A (1 )t a • (x)*^ + (i)i Q *A . (2.66) For layer (—1), we obtain A(_i)a Q/9 = A(o)V>, a ' (o)V,/3 + (0)¥>, a 'A(0)V^ - ( )/i~ (A (0 )¥> ia • (o)t,^ + A ( )i • ( )V ia + A ( )V i(3 • (0 )t Q + (o)¥>,/? * A (0) t „ ) - (A (0 )V3, q • (_!)* /J + A ( _ 1) t ) ^ • (0) <p Q + A (Q)^ • (_i)t a 22 + (0)V,/3 •A(_l)t, a ) + (0)^" (0)* a ' (-1)* jU + (0)*,a "A (_!)<: (/ 3 + A ( )*,/3 * (-1)*,q + ( )t,/3 *A(_i)t iQ ) + ( ( o)/l _ ) (A ( )£, q • (0)*,/3 + (0)*,a •A(o)t^) + ( ( _i)/i + ) 2 (A ( _i)t a • { _i)t ija + (-!)*« • A (-i)*^) , (2.67) A (_i)7 Q = A ( o)V jQ • + ( )¥>, a * A ( _!)t - (0) /r (A (0) t, Q ■ (_i)t + (0)*,a •A(-i) ,o • + (-yt^ »A(_i)t) , (2.68) A(_i)/C a /j == A (o)^ • (-l)*,/3 + (0)V, o ' A - (0)^~ (A (0)* a • {-1)*JS + ( )*,a •A^t./s) - (A ( _i)t a • (_Dt l/3 + ( _i)* Q •A(_ 1) t i8 ) . (2.69) Remark 2.3. The above (2.63)-(2.68) are for the incremental strain measures for sandwich shell. To obtain the variation of these strain measures, we simply use the same relations with A replaced by 8. I 2.3.5A Linearized contact weak form We now derive the linearization of the contact weak form, which requires the lin- earization of the resultant contact forces/couples. Substituting the one-parameter family of the perturbed configuration (2.47) into the static weak form (2.32), and then taking the directional derivative, we obtain DG C ($,5#)-A$ := ae (2.70) £=0 The complete linearization of the contact weak form G c ( 3> , <$<!> ) can be divided into two parts, the material part and the geometric part. We will discuss these two parts in detail in this section. To make the derivation simpler, we express the contact weak form in matrix- operator format. 2.3.6. Matrix-Operator Format of Contact Weak Form Let the material membrane force, shear force, and moment for layer (£) be defined as follows 23 (i)Jo v ' (2.71) M ~11 — 22 — 12 w m , (/) m , {e) m )'■ where is the Jacobian determinant in the material configuration evaluated at the cen- troidal surface of layer (£). We also define the director rotation matrix for layer (£) as follows A := and let A := , (£)*2 (<)An (<)Ai2 (<)Ai3 (£)A 2 l (£)A 2 2 (£)A 23 (/)A3i (^)A 32 (i)A 33 (£)An (<)Ai2 (f)A21 (€)A 2 2 (2.72) 3x3 (£)A3i (£)A 32 (2.73) 3x2 which simply represents the first two columns in qqA . From here on, we will not main- tain a rigorous difference in notation between tensors and the matrices of their components. Thus, bold-face symbols are also used to designate the matrices of components of tensors with respect to the spatial basis { e x , e 2 , e 3 }. With this understanding in mind, the ma- trices of components of the deformation map of the reference layer (0) and of the director for layer {£) are written as follows (o)¥> := (o)¥> (o)<£ 2 W t := (i)t (0 L (0)P J 3x1 while the variation and the increment of ( )V are written as t 2 t 3 (2.74) J 3x1 ^(o)V 1 <5(o)</? 2 <5(o)</> 3 A( )¥> 3x1 A ( )V 2 . A (o)¥> 3 (2.75) 3x1 For the variation and the increment of the layer director {t)t , we need to account for the no-drilling dofs condition. The matrices of components of 5 and A are 5 {e) t 2 L 5 (i) t3 A {e) t := 3x1 A ^t 1 A {e) t 2 A {e) t 3 (2.76) 3x1 24 The material counterparts of 8 ( ( )t and A y)t are respectively 8 W T and A W T , and are related to 8 and A by 8 m t = {e) A 8 {e) T, A {( )t = w A A W T . (2.77) The no-drilling-dof condition imposed on 8 (<) T and A W T is written as follows 8 m T -^3=0, and A (<) T • £ 3 - , (2.78) and thus if the matrices of components of 8 (£) T and of A W T (with respect to the material basis { Ej }) are defined as 8 {e) T = 5 ie) T* 2x1 A^T 1 a w t 2 2x1 (2.79) then (2.80) Also 8<f> := * (0)V *<-i)T S {0 )T L*d)T A$ := J 9x1 A (0) r A(i)T 9x1 (2.81) We now will obtain the operator expression of the weak form for each of the three layers. Since layer (0) is the reference layer, to which the two outer layers are referred to, we begin with layer (0). From the membrane part of (2.33), the expression for 8 (o)Q a /3 similar to that of A (o) aa/3 in (2.63), and using the symmetry of the membrane forces (o)^ Q/3 > we obtain ~a0 c o(0) n °(0) Oq/j -11 r i -22 r -(0)U (o)On + - (0)71 0(0)O 2 2 + \ ((0)" 12 5(0)Ol2 + (0)n 21 S (0)O21 ) d d = I ^T 5 (°)^ ] (o) n +1 (0)V>,2 ^77^(0) V J (0) n 22 + ^fS^^WV + (0)v5^r5(0)V»l (o)" 12 . (2.82) 25 Introducing the following operator for the membrane action in layer (0) . d (o) B mm we can then rewrite (2.82) as 1 (o)V>,i t d (0)V,2 t d , d (o)V,i + (o)V,2 (2.83) 3x3 (0) n a0 8 m\a n a = (O)<ta0 = (0)3 o (0) B mm 8 ( )V (0) N (2.84) Similarly, we introduce the following operators related to the bending and shear actions in layer (0): d (0)Bbm (0)*1 (0)Bbb t 9 (2.85) 3x3 t 9 (0)V,2 d (2.86) d 3x3 *5 MM • (O)-Dsm (0)1 ^e 1 ^e 2 (0)-Ds6 (o)V»;i (2.87) 2x3 2x3 Then following the similar procedure as described in (2.84), from (2.33), (2.63), (2.62), we obtain the operator format for the shear part and for the bending part of the weak form as follows - 1 (o)<T<5(o)7a = (o)Jo (o)B am 5 (o)ip + (o)B sb 6( )t (o)Q , (2.88) (0) h a0 8 t (O)K a — (Q)j (o)-Bftm <5 (0)<fi + (0)B bb 8 ( )i (0) M (2.89) 26 The contact weak form (2.33) for layer (0) can now be written as (0) G C (*,<**) = /{ A + t ~ (o) J5mm 8 {o)<P (o)N + (o)B sm 5 (o)<p + (o)B sb 5( )t (0 )Q (0)5^5(0)^ + ( )B bb 6 (0 )t] £ ( )M \ ( )j dA. (2.90) Remark 2.4. We refer to Simo and Fox [1989, eq. (6.25)], which is an expression similar to (2.90). I To obtain a simple representation for all three layers, we define the following gener- alized resultant force for layer (£) R := (e)Q M (2.91) 8x1 Recalling the relationship between 6 y)t and 8{f)T as given in (2.77), we combine the differential operators for membrane strain (2.83), for shear strain (2.86), and for curvatures (2.86), all for layer (0), into (o) B (0)°nm 3x3 03 X 3 3x3 {0)B sm 2X 3 (0)B s f, 2X 3 (0)Bbm 3x3 (o)Bbb 3X 3 (2.92) 8x12 where A is the director rotation matrix for all layers defined as (-D A A := (i) J (2.93) 12x9 Then, the contact weak form (2.90) for layer (0) can be written concisely as ( o)G c (*,<$#) = J {0) B6$* m R {0) ] dA. (2.94) For the membrane part of (2.33), the expression for S (_i)a Q/3 similar to that of A ( _i)a Q/? in (2.67), and using the symmetry of {-\)n al3 , we obtain ~a8 c ^ —11 r 1 ~22 r -(_!)n d ( _i)a a/3 = - ( _i)n d ( -i)a n + - ( _ 1} n d ( _ 1) a 22 27 + \ ((-i) n12 S (-i) a i2 + (_i)n 21 5 ( _i)a 2 i) <*(0)V,1 * (-1)^,1 - (-l)^ + <5 (-!)*,! * (-1)^,1 - ( )^~<5(o)*i * (-i)V 3 ,! + [<* (0)^,2 * (-1)¥>,2 - (-1)^ + 5 (-1)*,2 * (-1)^,2 - (0)^ <5 (0)*,2 * (-1)V,2 + (_ 1} n 12 [<5 (0)^,1 * (-1)V,2 + 5 (0)V, 2 ' (-1)^,1 - * (-1)^,2 (_l)/l + (J(_l)t2 * (-1)^,1 - (0)h 5 ( )*,i * (-1)V,2 ~ (0)/i <5(0)*,2 * (-1)V>,1 (2.95) Upon introducing the following operator (-i)-B memb associated with the membrane action in layer (—1) (— l)-*-*memb • — . d (-*>¥> i (-^,2^1 05x3 (-1)^*1 5 . _ « d - {0 )h ( _ 1)¥>1 _ 1x3 (-1)^2^ 1x3 5x3 (o)h~ (-1)11! lx3 05x3 05 X 3 (2.96) where the operator r-nlli is defined as A 9 d (-1)11! := ( _ 1)V > 2 _ + i-d^ , (2.97) we can rewrite the membrane part for layer (-1) in (2.33) in a compact format as follows -(_i ) n Q/3 ^ ( _ 1) a a/3 = (_i)B memb <5$- . (2.98) From (2.68), we define the differential operator associated with the shear action in layer (—1) as follows (-i)B shear 28 3 x3 . d dp 03x3 3 x3 t ® (-D^,i - (-1)" (-1)* (o)h ( _i)t 3x3 o 3x3 d 1x3 (-1)^2 - (-1)* i 9 dp (o)h (-i)t 03x3 0*1 03x3 3x3 1x3 (2.99) Similarly, from (2.69), we define the differential operator associated with the bending ac- tion in layer (—1) as follows (-l)Bbend '■ = o 5x3 05x3 05x 3 TT 9 - (0)^ _ (-1)*, 1 9^ J TT 9 - (o)h~ (-i)t 2 d ' dp - (0)/l~ (- 1)114 1x3 1x3 1x3 A , (2.100) where the operators (_i)II/, for / = 2 , 3 , 4, are defined as (-1)112 := (-i)^i - (-i)^ + (-1)*, i i (-1)113 := (-i)¥>' 2 _ (-i)^ + (-1)*,2 i t d (2.101) (-i) n 4 : - (-1)^2 ^tt + (-i)*!i dp dp ' We can easily verify that the shear part and the bending part for layer (-1) in the weak form can be written in a compact format as follows (-1)9°" <*(-l)7a = (-l)B s hear<5* * (-l)-R, (2.102) 29 (_l ) m Q/3 ^(_i ) /C Q/3 = { -i)B hend 8$ ' (-l)-R Now let the combined differential operator for layer (—1) be (-l)-B : — (-l)-Omemb T (-1)-° shear T (-l)-Dbend -Bmemb + f-ll-Bshear + t-l)By From (2.96), (2.99), and (2.100), we obtain B = j 9 (-DV,i t d (-i)II, t 5 (-«* ae" (-1)^,2 (-»« ^2~ - (-i)* (-1)^2 - (-1)* (-D /i+ (-l) n i 5 (-l) n 2 (-l) n 3 g (-1)114 TT 5 TT 5 (-1)112^- +(-!) n 3^T We thus obtain the following compact expression 1 ( o)/r (-i)v; 2 ^i- u ix3 - (Q)h~ (-i)Hi lx3 5 - (o)h (-i)t l (o)h { -i)t (0)h (-1)*,2^72 Ulx3 d (2.103) (2.104) 0, 1x3 1x3 t d (0 )ft (-i)t,i ^-f lx3 0, (o)^ (-i)n 4 o lx3 (2.105) - ( _ 1 )n Q/3 5 ( _ 1) a Q/3 + ( _ 1) 9 Q 5 ( _ 1) 7 Q + ( _ 1) m^5 ( _ 1) « Q/ 3 = ( _ 1} £ 5 $ • { _ 1} R , (2.106) and the contact weak form of layer (—1) as (2.107) For the top layer (1), similar to the definition of (-i)J3 for layer (—1), we define (i) B := d d (DVfi °ix3 (o) h+ tiMi 7*71 d d (i)^2^2 01x3 m h+ (0)h + (1)11! (1)11! 0i x3 . d (i) 1 d£ 2 0lx3 (0) h + (1) t* a 1x3 (0) 0^ 0^ J? d T^TT ° lx 3 (o)^ + (l)*4 (1)*,2 df 2 (i)Il 6 lx3 1x3 (0) h+ {1) t- 2 t d (o)h + (i)II 6 w 71 (i)^Ii ^71 (i)*"(i) n i (l) n 2 (l) n 3 n 9 d (1)114 d (i) n 5^FT +(i) n 4 d 30 (2.108) where the operator (1)11/, for 1=1, 6, are defined as follows tt t & t ® m lli := ( i)¥> )2 ^j- + (i)V>,i , (l) n 2 (l) n 3 M<P,i + (i)h (i)t — , (1)^2 + d) <£ ^" . (1)114 := (1)^1 + (i)*,! , (XjIIg := (i)V>f 2 + (1)*,2 , > t 3 . d n fi := (1)116 ae 2 (2.109) 31 The contact weak form of layer (1) is then (1) G C (*,<$*) = J (1) B6*- (1) R {1) j dA. (2.110) A 2.3.6.1. Material tangent operator The material part of the tangent stiffness operator, denoted by DmG*A$ , arises as a result of linearizing the resultant forces/couples at a fixed configuration. We now treat each layer separately, as we did for sandwich beams in Vu-Quoc and Deng [1995]. Here, we only consider hyperelastic materials. Let tp be the energy function of the shell. We have the following constitutive relation (Simo and Fox [1989]) dip _ dip _ dip (e)n a0 = (i)p , {l) q a = {t)P « , , {i)m a0 = {t) p- — — . (2.111) O(e)£a0 O {l)Oa O (t) Pa0 For each layer, we have 2> W R«A$ = (i)C( f )BA$ , (2.112) where the tensor (^C of elastic moduli is given below V CN <0 % 2 (N CO n T co •^crT cn CO H Qi co" -9co" %> els' •&CO CO - CO « CN ^ -I to co" CO Q. CO - CN CO CO CO CN «0 c6" co" cn CN 54. els' CO' •3 CO" CN CO -9- — cn CO CO - CO" cn CO iH CO" V ft tN CN CN CO CN Ci. a co" -^.CO CN H CN M "2 -3.CO CN CN CN CO CN CO CO co" •3- co CM * co" -5- CO CN CO -=>.co CN " 05 £ 3 CN CO * CO - CO c-o CN CO CO CO CN CN ^> co" ■3- co CN co 2 -9- CN CO CO CO CO CO -5H - CO CO cn «0 c$* 9 CN CO CO tN tN c6" CN CO CO CN co" CN CO -3co tN CO M to co" •3 - IN co CN cb -3). CN CO CO co" CO CN C4 ^* co" %3- CN CO CO CN s CO" "9- co CN ^ CO 2 cu ■3 =• CN CO -9- ~ CN CO CO CN CO CN CO tN CO o l<5- 33 For hyperelastic materials, the elastic moduli for the membrane, shear, and bending actions for layer (£) are given as in (2.23)-(2.25). The matrix of the (tangent) elastic moduli in this case takes the following form (*)C := sym. with (£)C m := {e) c% (t)C s := sym. /->33 03x2 03x3 (e)C s 02x3 (e)C b := 3x3 (2.114) 8x8 ■>11 sym. , x/ n '22 3x3 (2.115) 2x2 The coefficients of the above matrices are given below, for the membrane action: (e)E {e) H ^13 /o22 r<23 /o33 1 - («)f) W E W H 2 W^ 11 M^ 11 . »'»i4 11 W A«+(l-(o«') W ^ 1, W ^ U ]. 1 - (w^) 1- ((/)»/) 2 W^ 1 (*) /l22 4 22 2 W A (i) A (2.116) 22 /i 12 2 tt)A tf)*- , 1 - w W A + — - — {e) A (e) 12 „ A n 34 For the bending action: d)E {e WW C 13 12 (l - {i)E {l 12 (l - CP 12 (l- 12(1 - (i)E {l CP W^b 12(1 - 12(1 - For the shearing action: # 3 A n A 11 0" tf 3 # 3 H 3 H 3 H 3 (0 v {e) A n {e) A 22 + (l - m v) {l) A 12 {e) A 12 } , All Al2 {I) A ( t )A , (t)A 22 {e) A 22 , (2.117) A 22 4 12 l ~ M V 4" ./I22 , 1 + (t) V 4 12 Al3 {t)A (e)A H f^/l f^A W^s 11 = W«s (<)A U , (()Cl 2 — [i) K » {t)G (i)H ( ( )A 12 , C 22 = (t)K s (£)G (t)H (t)A 22 The tangent material stiffness operator for layer (£) is thus V M[i) G c (#,J#)-A# = I [ W B*$« (<) C W BA$ 2.3.6.2. Geometric tangent operator (2.118) dA. (2.119) The geometric part of the tangent stiffness operator, denoted by D G G*A&, arises from the linearization of the geometric part of the contact weak form, while keeping the material resultant forces/couples constant. We now treat each of the three layers separately, as we did in dealing with the material stiffness operator. Remark 2.5. It is noted that while the principal kinematic unknowns are mtp and [fit, for I = -1,0, 1, the computational kinematic unknowns are $)(p, , for I = 35 -1,0, 1, where mO represent the rotation vectors that rotates E$ to {e) t at the current state at time t. It is important to note that {i) 9 does not represent the time history of the motion of the director {l) t , but only relates the directors between the material configuration and the current configuration. In the linearization procedures, the primary variables (i.e., the variations to be held constant in the linearization process) are S {t) instead of 5 {e) t , which we will explain in Remark 2.6. We recall that || {g) t ||= 1 =>6 {t) := [^t x 5 {e) t, or 6 {i) t = 5 {t) x (t) t . (2.120) we thus obtain the increment of the 5 from (2.58) as A(<S ( /)i) =5 {e) 0xA {e) t = ( w l x5 w t)xA {e) t = - (A (<) t -5 w t) (e) t , (2.121) I Remark 2.6. From (4.19) and (4.12b) of Simo and Fox [1989], (84) 3 and (156) of Vu-Quoc et al. [1997], the equation of balance of angular momentum for a single-layer and also (A {e) t'S {e) t) {t) t a . (2.122) shell is Jt (2.123) Alternatively, it also can be written in the following form 1 (2.124) Jt Since II t 11= 1, we differentiate it twice to obtain t ' t l=>ft=0=$>t't + t't = 0. (2.125) 36 Thus, t _L t , let « be define as w := t x t . (2.126) We obtain u = txt + txt=txt. (2.127) Since (a? x t)^t*[w,t,Jt]*w»(t X<5*) « ut'69 , (2.128) w x t = x V J X t (*•*) t - (* •*) * = *+ II * f * . (2.129) and also since t *5t = 0, we obtain (w x t) -6t = , (2.130) so that Tp'i'St = 1 p u '89 . (2.131) In the above equation, I p u> corresponds to the inertia term on the right hand side of (2.123) (i.e., the balance of angular momentum in terms of the weighted resultant moment fh a ), whereas I p u corresponds to the inertia term of (2. 1 24) (as a result of (2. 1 27)) (i.e., the balance of angular momentum in terms of the physical resultant moment m a ). We now point out that the reason to use 8 , instead of 5 1 , as primary variable: The use 5 9 is more convenient. Note that i ^ w x t . (2.132) Since t • t = and (2.126), we obtain t-wxi, (2.133) u> x t = ijj x (uj x t] = (wt\ w - (wu>) t = - || uj || 2 l»t , (2.134) 37 Making the derivative of t = u> x t , we obtain ** • * • n2 t — IjJ X £ -f U> X t = u x t — II UJ 11 1 • I (2.135) I For the reference layer (0), in (2.94), we hold the resultant forces/couples ( )-R fixed, and linearize the geometric part, that is, we are finding the expression [P( (0 )B5*) *A$] '(o)R , (2.136) in operator form, where the operator ( 0) B was given in (2.92). From (2.92) and (2.81), (2.136) can be expressed as X>( (0 )B(J*) • A* •«,)•** = {v[ {0) B mm 6 {0) (p] *A (0 )#}'(o)iV [i] + [V [(o)B m S (0 )V + (o)£ s6 <5 (0 )T] •A(o)*} • (0 )Q [2] + {D [wBbmSwfp + i0) B bb S [0) T] .A (0) *}-(o)M, (2.137) [3] where the differential operator (o)B m , (o)B sm , $)B*ni (o)B sb ,and ( )B bb were given in (2.83), (2.86), and (2.87), respectively. We now proceed to give a detailed expression for part [1] in (2.137). From (2.83), we obtain [1] = A(o)V'i (o)V,2 + A(o)^'i ^(o)V,i . (0)TI —22 (0)? ? (o)» 12 [A ( o)^! •5ffl)(fi 1 (o)n 11 + A (0 )Vj *<5(0)V,2 (o)« : + { A (o) ( P\ '^(0)^,2 + A (o)^i (o)V,i ) (o)^ 12 xx22 (2.138) 38 We define the geometric differential operator for layer (0) as d (0) T := 03x3 3x3 o 03x3 3x3 03x3 3x3 1 3 03x3 3x3 1 3 3X 3 3x3 Is 3 ^3x3 d d 3x3 o 3x3 A 12x9 (2.139) 15x12 and the tangent geometric moduli for the membrane action in layer (£), for £ = -1, 0, 1, as (2.140) {e) n lL 1 3 {e) n LZ 1 3 3x9 W" 12 Is (e)n 22 1 3 3x9 0g x3 9x3 9x9 It is easy to verify that [1} = {v[ {0) B m 5 {0) <p} •A (0) #} « (0) JV = (0) Ti$' (0) JfJJ (0) TA$ . (2.141) Now, for Part [2] in (2.137), from (2.86), we obtain [2] = \v [(o)B sm 6 {0 )<P + (o)B sb 5 {0 )t »A(o)$ } • (0 )Q = [A ( oy**-<J ( o>V>,i (o)? 1 + A (0 )i ( *6 i0) <p t2 (0) 9 2 +A (0) ¥>; 1 -5 {0) t ^q 1 ~ (<Q)V>fi (o)*) (^(ojt »A(o)t) (o)g 1 + A (o)^ «<5( )t (o)<7 2 - ((0)^2 (o)*) (<5(o)* *A (0) t) ( o)g 2 ] • (2.142) Remark 2.7. The matrix (0) T in (2. 1 39) has five rows of submatrices and four columns of submatrices. The four columns of submatrices correspond to {0) <p, { _ 1} t , {0 )t , mt ac- cording to the ordering in 5 $ (See (2.81)), and 5 {e) t = {e) A 6 {l) T (See (2.80)). I 39 Let the tangent geometric moduli for the shearing action in layer (£), for I = — 1, 0, 1, be defined as follows Tfl2 3 x3 3x3 I3 3 x3 3 x3 (£)Q J-3 WQ 2 Is - (09° (<)7a I3 06x3 06x3 06x3 (2.143) It can be verified that (2.142) can be written as [2] = {p[(o)B sm £ (0 )¥> + (o)B sb 5 {Q) t] 'A(o)*} • (0 )Q (o)T^' ( o)4 2 (o)TfA$ (2.144) For part [3] in (2.137), from (2.87), we obtain [3] = {V [ (0)-B6m^(0)V + (0)-B 66 (5(o)tj 'A( )f }'(0)M = [( A (o)*?i ^(o)V,i +A( )V»; 1 *^(o)*i + (0)¥>a •A^<o)*,i) (o)^ U + (A ( )t* 2 '(5(0)^,2 + A (0)^2* 5 (0)*, 2 + (0)^2 * A5 (0)* 2 ) (0)^ + ( A (0)*fl (0) V,2 + A (0)^2 (0)V,1 + A (0)^i *6 (0 )t,2 +A (0 )vJ 2 ^(o)*,i + (0)V»*i •A<5( )t,2 + (o)^2 *M m t,i) (o)m 12 } ■ (2.145) Let the tangent geometric moduli for the bending action in layer (I), for £ — —1, 0, 1, be defined as follows 3 x3 3x3 3x3 (e)rn 11 1 3 w m 12 1 3 o 3x3 o 3x3 o 3x3 w m 12 1 3 (e) m 22 1 3 o 3x3 o 3x3 - (e)m la (/)7a 13 - w m 2a W 7 Q 1 3 (t)m n 1 3 {i)rn n 1 3 - (e)rn la {( )la h 3x3 o 3x3 {e) m 21 1 3 (t) m 22 1 3 - (i)rn 2a W 7 a 1 3 o 3x3 o 3x3 (2.146) Using Remark (2.5), we can verify that [3] = |X> [^BbmS ( 0) (f + (0)^66^(0)* 'A ( o)^} (0) M (2.147) 40 Let the geometric stiffness moduli for each of the three layers be defined as r ~ii (0 n J-3 =12 i (0 n 13 . /rl 1 (0 m X 3 (0 m L 9 {Qtl 1 3 (£)n 22 1 3 —2 1 w m 12 1 3 (e) m 22 1 3 la (0<7 x 3 - w m lQ (£)% 1 3 - (0^" 2Q (<)7a 13 (i)rn n u w m 12 1 3 ™ ic* - (0 m (07a 1 3 03x3 03x3 — 21 . (t)m h {e) m 22 1 3 - (f)m (07a 13 03x3 3 x3 (2.148) where {e) C := - (07a + (e)m a/3 «)««„) . (2.149) Adding up (2.141), (2.144), and (2.147), we obtain the following expression for (2.137) {0) R = (0 )T<5*»JC£ ( o)TA$ . (2.150) [d( (0 )B(5$) 'A$ The tangent geometric stiffness operator for layer (0) can thus be written as Do<oA (♦.,**) -A» = | [(fl)T^- ( o)^(o)TA$] ( o)7o^- (2-151) A For the bottom layer (—1), in (2.107), we now hold the resultant forces/couples (-i)-R fixed, and linearize the geometric part, that is, we find the expression for »A#] •(-i)Jl (2.152) 41 in operator form, where the operator (-i)-B was given in (2.105). We define the two differential operators (-i)Ti , and (-i)T 2 for the bottom layer (—1) as follows (-i)Ti := V 1 -(-1) d - (o)h~ i 3 9 3 x3 -is 3 x3 03x3 03x3 03x3 1 a 03x3 " o 3x3 13 3 x3 3 x3 " 3x3 l3 ae 3 x3 3 x3 3 x3 3 ae 2 03x3 3x3 T 2 := 03x3 03x3 13 3X 3 3 x3 3 x3 03x3 03x3 03x3 13 o e 03x3 3 x3 3 x3 03x3 3 x3 12x9 j 15x12 (2.153) 12x9 (2.154) 18x12 In addition to the tangent geometric moduli (-\)K l G that corresponds to the bottom layer (-1) as an independent single-layer shell, we have also the tangent geometric moduli (-i)-Kg tnat comes from the coupling between the bottom layer (-1) and the reference layer (0). The tangent geometric moduli {-i)K% for layer (-1) can be written as follows 42 (-l)-K'l 13 {-\)K 2 lz (-1)^3 13 {-1)^2 I3 O3X3 03x3 0g x g (_l)i^3 1 3 3X 3 3X 3 (-1)^4 1 3 (-l)-^5 13 (-i)K e 1 3 (-1)^5 I3 03x3 3 x3 (-i)K & 1 3 3 x3 3 x3 (2.155) The parameters in the above moduli matrix ^i)Kq are : = {-i)h + ((-D^ 11 (-1)^1 + (-i)^ 12 (-1M2) (-i)*.i + (-i)h + ((-i)n 22 (-i)vfa + (-i)" 12 (-i)Va) (-i)*,2 + ((-i)m 11 (_i)t, 1 • 1 + (-i)m 22 2 * (-1)^,2) +2(_!)/i + (-i)m 12 • (-i)*,2 1 (_D^ 2 := ((-i)« n (-1)^1 + (-ip 12 (~1)<P%) (-i)t + (-i)^ + , (-l)#3 — ((-i)^ 22 (-1)^,2 + (-D« 12 (-1)^1 ) (-1)* + (-D^ + (-i)Q 2 . (-i)K 4 := (0) /r ((-i)« n (-i)Vj + (-i)" 12 (-1)^2) (o)*,i + (0)h" ((-i)« 22 (-i)<p% + (-i)^ 12 (-l)V'l) (0)*,2 + (0)/l~ ((-l)? 1 (0)*, 1 + (-l)Q 2 (0)*,2) ' (-1)* + ( )/i~ ((-i)fn 11 • (o)t,i + (-i)"^ 22 (-i)*2 ' (0)^,2) + (o)^~ (-i)™ 12 ((-i)*,2 * (0)*,i + (-l)*l * (0)^,2) , {-i) K s := (o)h~ ((-i)n 11 (_!)V»fi + (-i)™ 12 (-1)^2) (0)* + (0)^" (-i)^ 1 (0)* * (-1)* 43 + ( )/T ((-i)rn 11 (_i)tj • (0 )t + (-i)m 12 ( _i)t 2 * (o)*J . ( _i)K 6 := { )h~ ((-i)n 22 (_i)^2 + (-i)" 12 (-D^i ) (o)* + (o)h~ (-i)Q^ (-1)* ' (o)t + (o)h~ ((-i)rn 22 (-i)*,2 * (0)* + (-i)"^ 12 (-i)*.i * (o)*) • The tangent geometric stiffness operator for layer (-1) can thus be written as V G{ -i)G c (#,*$)«A# = J [ (il )T 1 5¥«(_i ) ICj; (-dTjA^ .4 + { _ 1) T 2 5$- ( _ 1)J ft: 2 7 ( _dT 2 a*] (-dJo^- For the top layer (1), in (2. 1 10), we now try to obtain the expression (2.156) (2.157) (2.158) [P( (1) B<5*) -A* in operator form, where the operator (i)B was given in (2.108). Similar to (2.157), we obtain the following tangent geometric stiffness operator for layer (1) V G[l) G c (■#,*•) -A$ = / ■ x {T l 6*' {x) K l G ( i)TiAl + (1) T 2 <5*' ( i)^G (i)T 2 A*| (1) j dA, (2.159) where (i) T i d d o 3x3 (0)^ + J-3 d 3x3 (0)^ + J-3 ae 1 (i)^ 13 d (i)h 1 3 9x9 d i 3 d A 12x9 . (2.160) 15x12 44 (1)T 2 := 03x3 03x3 3x3 03x3 3x3 1 3 3x3 d d 03x3 3x3 ^3^2" ^ 3x3 03x3 3x3 03x3 3x3 3x3 1 3 03x3 I3 d 0,x3 3x3 u 3x3 3X 3 I3 d 12x9 18x12 1 3 (1)^2 I3 (1)^3 h 1 3 3 x3 3X 3 (i)K 3 1 3 3 x3 03x3 9x9 9x9 1 3 (1)^5 I3 h (i)K s 1 3 3 x3 3 x3 (i)K e I3 3 x3 3 x3 (2.161) 18x18 (2.162) The parameters in the matrix ix)K 2 c are defined as follows ~(o)h + ((i)n 11 (i)(p\ + (1) n 12 (i)^) (0) t, ~(0)h + ((i)n 22 (1) v>f 2 + (ijn 12 (1)^1) (o)*2 ~(o)h + ((i)^ 1 ( o)i, 1 + (0 )i 2 ) * (i)* ~(0)h + ((ijm 11 {!>*,! • ( o)t 1 + ( i)m 22 (1) i 2 ■ (0 )t 2 ) 45 -(o)h + (i)rh 12 ((i)*,2 * (o)*,i + (i)*i * (o)*,2 ) ) (i)K 2 := ~(o)h + ((i)n n ( i)^j + ( i)n 12 (i)<^ 2 ) (0)* - (o)h + (i)? 1 (i)t • {0) t ~(0)h + ((i)m n ( i)t,i • (0 )t + (i)m 12 (1) t 2 ' (o)*) ■ (1)^3 := ~(o)h + ((l)" 22 (i)Vf 2 + 0)™ 12 (i)^fl) (0)* ~ (o)h + (i)9 2 (l)* * (o)* - {Q )h + ((i)m 22 ( i)t 2 * (o)* + (i)^ 12 • (0 )t) , (i)^4 := (d)" 11 m<P*i + (i)" 12 (1)^2) - W h ~ ((D^ 22 (D^2 + (i)« 12 (i)<P*i) (i)*2 - (1)^" ((l)™" (i)*i •«*,! + (i)"^ 22 (i)*2 ■ (i)*,2 - 2 (1) rn 12 1 • (iy*,a) , (1) ^ 5 := ~m h ~ ((I)"" WVfi + (!)" 12 (1)^2) (i)* ~ W h ~ (i)^ ' m Kt ■= -(o) h ~ ((i)^ 22 (1)^2 + (i)^ 12 (i)Vfi ) (i)* - (i)^" (1)9* • (2.163) Remark 2.8. Even though (\)K l G has the same form as (o)K G (i.e., the operator of a stand-alone layer (1)), the operator (1) Yj has the coupling terms in the submatrices (1,3), (1,4), (2,3), (2,4). In of (2.160), there are five rows of submatrices and four columns of submatrices. The four columns correspond to ( )V?, (-1)*, (o)*» (l)*- The coupling terms in (1 ) Y x , when multiplied with A ( )t and A { i)t in ( A A <*> ), only affect the submatrices (1,1), (1,2), (2,1), (2,2) related to the membrane forces {e) h n , w n 12 , (t)n 21 , (e)n 22 . The reason is the offset of layer (1) with respect to reference surface, which is the centroidal surface of layer (0). Another way to understand (i)Yi (or the meaning of the coupling terms in (1) Yj , i.e., the difference between (0 ) Y and (1) Yi ) is to think of layer (1) as a stand-alone layer, initially at the same location as that of layer (0). Then (i)Ti would be similar to ( ) Y (note the difference between the column corresponding to in (i)Yi and the column corresponding to ( )t in (0) Y ). Now shift layer (1) to the top of layer (0); then the mem- brane forces in layer (1) must generate some additional moments. The coupling terms in 46 M T, Play the role of lever arms. On the other hand, in (1) K 2 G , we have all the coupling I terms, but not in (i)"T 2 . 2.4. Numerical Examples for Statics of Sandwich Shells The finite element formulation for the statics of geometrically-exact sandwich shells presented in the previous sections has been implemented in the Finite Element Analysis Program (FEAP), developed by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a DEC ALPHA with the DEC UNIX V3.2D-1 operating system. Linear finite-element basis functions are used in the examples in this section. To avoid shear locking, selective reduced integration is used to evaluate the shear part of the tangent material stiffness matrix {l) K M and the tangent geometric stiffness matrix m K and also the shear part of the residual force matrix, whereas the bending part and the membrane part of the tangent stiffness matrix and of the tangent residual force matrix are evaluated using full integration. To identify the correctness of the present theory and the related coding, we tested several examples of sandwich plate with different aspect ratio. The aspect ratio is defined as A := i . W ■ T , where (L, W, T) designate the length, width, and thickness of the sandwich plate, respectively. For the bending stiffness and membrane stiffness, we use the full 2 x 2 integration points, whereas for the shear stiffness, we use the reduced 1 x 1 integration point. Firstly we tested the bending of a sandwich plate with three identical layers employ- ing 90 four-node quadrilateral elements. The plate is clamped at the edge = 0, and is free at all the other edges (See Figure 2.2). The vertical displacements and the tip rotations are compared with the theoretical results and with the results obtained from the single- layer theory. The results, with a maximum error less than 0.4%, also show the ability of our sandwich shell elements to model the anticlastic curvature. We also consider the Cook problem with only the core layer using 100 sandwich shell elements. The results agree well with those of single-layer shells (Rifai [1993, p. 173]). To demonstrate the ability of the formulation to capture large rotations and displace- 47 e 3 = E *12 h < /i i 12 ^ (0)^ )<-l) m .12 Figure 2.2. Sandwich shell with three identical layers ments, we tested the torsion of a cantilever plate with three layers using 20 sandwich shell elements (Figure 2.3). To justify the computed results, we use the theoretical rotation of the torsion of an elastic bar given by 6 = TL/GJ k as a basis for comparison, 3 where L is the length of the plate, G is the shear stiffness and J k is the polar moment of inertia of the plate cross section about the centroidal axis along the length of the plate. Since the direction of the concentrated forces remains fixed along the £ 3 axis, the resulting couple T generated by these forces decreases with the twisting of the bar (See Figure 2.4), as the dis- tance d between these forces becomes smaller. Thus to make the comparison meaningful, we use the final value of the couple at the last time step as the torque T used to compute the theoretical rotation 9. The difference between the theoretical results and our FE results on the twisted angle is 9%. Figure 2.3. Torsion of a cantilever plate. 3 Roark and Young [1975, p.290] presented the formulation to calculate the angle of twist for beam with solid rectangular sectio°n, which gives very close results when compared to the above formulation « 0.2%). 48 £ 3 d \ F e 0.2 Figure 2.4. Force couple, at the tip of the plate, generated by a couple of concentrated forces. 2.4. 1 . Roll-down Maneuver of a Sandwich Plate We now consider the roll-down maneuver of sandwich plates. First, we tested the sandwich shell having only the core layer using 10 elements. Comparing to the theoretical deformed shape (i.e., a full cylinder), the relative error in the tip displacements is 0.4% in the f 1 direction and 0.005% in the £ 3 direction. The displacements obtained with the sandwich shell code are exactly the same as those with the single-layer shell code. We also tested the roll-down of a sandwich plate with only one outer layer. We still obtained good results even with the slower rate of convergence. Next, we consider the sandwich plates with three identical layers. The material prop- erties and geometric properties are chosen as follows: = 1.2 x 10 7 , w i/=0.0, w « s = 0.75, w h = 0.033333, for £=-1,0,1. where , , ^k s are the Young's modulus, the Poisson ratio, and the shear correction coefficient of layer (£), respectively. The geometrical dimensions of the plate are length L = 10, width W = 0.1. At first, we use 10 uniformly distributed sandwich shell elements to discretize the (2.164) 49 sandwich plate. The computed tip displacements at the end of the last loading step are re- ported in Table 2. 1 . The computed displacement u 1 differs from the exact solution by 7.5%. In the first loading step, convergence is achieved after 10 iterations; in the last loading step, convergence is achieved after 1 1 iterations. Then we use 20 uniformly distributed sandwich shell elements to discretize the sandwich plate. The computed tip displacements at the free edge are reported in Table 2.1, where it can be seen that the displacement u 1 is closer to the exact solution of (-10) when the plate becomes a full cylinder. The relative error in u 1 is now 1%. Convergence is obtained after 9 iterations in both the first loading step and in the last loading step. Finally, the computed displacements at the free edge using 40 uniformly distributed elements shown in Table 2.1 are clearly closer to the exact solution, in which the value of u 1 should be (—10), and the value of u 3 should be zero. The relative error in the displacement u 1 is now 0.53%. Convergence is obtained after 10 iterations in the first loading step, and 11 iterations in the last loading step. Table 2.1. Roll-down of a sandwich plate with identical layers: Displacements of a corner of the free edge. Disp. 10 elements 20 elements 40 elements v> -9.25472 -9.81873 -9.94730 u 2 -3.08237 x 10" 11 -2.15727 x lO -10 -1.02421 x 10~ 9 u 3 -1.98609 x 10" 1 -1.05095 x lO" 2 -8.77204 x 10~ 4 Figure 2.5 shows the undeformed and deformed shapes of the sandwich plate using 40 sandwich elements. 2.4.2. Sandwich Plate with Ply Drop-offs In this section, we present the computational results for sandwich plates that have discontinuities due to disparities in the length of the layers, resulting in the so-called ply drop-offs. 2.4.2.1. Sandwich plate with ply drop-off We now consider a cantilever sandwich plate with three layers, and with a ply drop- off in the top layer at mid length, see Figure 2.6. The free edge at the tip is subjected to 50 Figure 2.5. Roll-down of a sandwich plate with three identical layers: Isometric view of deformed shape. a uniformly distributed force of n* 13 = 100. The geometric and material properties are listed below: L = 10, W = 0.1, T b = 0.3, T a = 0.2, (2.165) where L is the length, W the width, T b the thickness before the ply drop-off, and T a the thickness after the ply drop-off, and w £=1.2xl0 7 , (<)f=0., (£) k s = 0.75, for £ = -1,0,1. (2.166) Before the ply drop-off, the layer thicknesses are {e) h=0.l, for £ = -1,0,1. (2.167) After the ply drop-off, the layer thicknesses are ( _D/i = 0.1, {0) h = 0.1, {l) h = 0. (2.168) Since the plate has a large aspect ratio, we use the Euler-Bernoulli beam theory to predict the deflection. The bending stiffness coefficients of the beam before and after the 51 A A A / B L/2 L a - L/2 Figure 2.6. Sandwich plate with one ply drop-off. ply drop-off are £/ b = £W(T 6 ) 3 /12 = 2700, J5/ a = LW(T a ) 3 /12 = 800, (2.169) respectively. Let P = n* 13 W and M = PL a be the resultant tip load, and the internal moment at the ply drop-off. Let u\ be the transverse displacement at the ply drop-off B due to the force P and the moment M. Let u\ be the transverse displacement at C of thin half of the plate, with the section B at the ply drop-off clamped. The total transverse displacement u 3 at the plate tip is the sum of u\ , the transverse displacement at C due to the rotation B b of the section at B (this rotation results from the bending of the portion AB), and u\ : u 3 = ul + L a 6 b + u\ = {PL b 3 /3EI b + ML b 2 /2EI b ) + (PL b 2 /2EI b + ML b /EI b )L a + PL 3 a /3EI a = 1.6010. ( 2 - 17 °) Table 2.2 presents the computed results using 20 uniformly distributed linear sand- wich shell elements and using 20 uniformly distributed equivalent linear single-layer shell elements, respectively. The transverse displacement u 3 obtained with sandwich shell ele- ments has a relative error of 0.030% compared with the analytical result based on Euler- Bernoulli beam theory, and a relative error of 0.033% compared with the computed result 52 using single-layer shell elements. Table 2.2. Sandwich plate with ply drop-off: Tip displacements Disp. sandwich elements singer-layer elements v> -1.71076 x lO -1 -1.62236 x lO" 1 u 2 -1.43540 x lO" 13 3.58955 x 10~ 13 u:'' 1.55253 1.54842 Remark 2.9. Similar to the case of sandwich beams in Vu-Quoc and Deng [1995], the result with single-layer shell elements is smaller than that with sandwich shell ele- ments, since the equivalent single-layer plate has a symmetric ply drop-off, unlike the non-symmetric ply drop-off in sandwich plate. Further, hinge is not allowed to form in the cross section at the ply drop-off in the equivalent single-layer shell. Figure 2.7 depicts the undeformed shape and the deformed shape; the effect of the ply drop-off is not easily discernible. I Figure 2.7. Sandwich plate with ply drop-off subjected to tip force: Isometric view of undeformed and deformed shapes. 4 For moderate thick plate and Poisson's ratio v = 0, Euler-Bernoulli beam theory gives accurate results on displacements. 53 2.4.2.2. Two-layer plate with ply drop-off: aspect ratio A = 5 : 1 : (1, 0.5) Here the aspect ratio of the two-layer plate with ply drop-off is represented by A :— L : W : (7},, T a ). We now consider a cantilever sandwich plate with only two outer layers (and with the core layer inexistent), subjected to the uniformly distributed force of n* 13 — 60 assigned at the free edge. The plate has a ply drop-off at mid length. The geometric dimensions of the plate are (Figure 2.8) L = 5, W = l, 7; = 1, T a = 0.5, (2.171) with the layer thickness before the ply drop-off being ( _i)fc=0.5, ( o)^=0.0, (1) /*= 0.5, (2.172) and after the ply drop-off (_!)/» =0.5, (0) /i=0.0, (1) h=0.0. (2.173) The material properties chosen are (/)£?= 29,000, {l) u = 0.294, (0 « s = 1, for £=-1,0,1. (2.174) Ten uniformly distributed elements are used in the computation. The displacements of a corner node at the tip are tabulated in Table 2.3. The undeformed and the deformed shapes are shown in Figure 2.9, where a change in curvature at the ply drop-off is clearly discernible. Table 2.3. Two-layer plate with ply drop-off, A = 5 : 1 ; (1, 0.5): Tip displacements. u 1 u 2 u 3 -5.18243 x 10- 1 -4.22720 x 10- 4 1.55017 2.4.2.3. Two-layer plate with ply drop-off: aspect ratio A = 20 : 1 : (1, 0.5) We now consider a cantilever sandwich plate with only two outer layers (and with the core layer inexistent), subjected to a tip moment. The plate has a ply drop-off at mid length 54 L Figure 2.8. Two-layer plate with ply drop-off: Geometry and assigned force. Figure 2.9. Two-layer plate with ply drop-off. Aspect ratio A == 5 : 1 : (1, 0.5): Isometric view of undeformed and deformed shapes. (see Figure 2.10). Since the connection between the thinner part L a and the thicker part L b of the plate is flexible, the actual moment needed to bend the thinner part L a into a full circle is a little smaller than the theoretical result of M = 2irEI a /L a , which is obtained for an equivalent beam with a clamped end. The geometric dimensions of the plate are as follows L = 20, W = l, L b = 10, L a = 10, T 6 = l, T a = 0.5. (2.175) The layer thickness before the ply drop-off are = 0.5, ( )/i = 0.0, (i)h = 0.5 (2.176) 55 and after the ply drop-off =0.5, (0) A = 0.0, {l) h=Q.O (2.177) The material properties chosen are {t) E = 29000, {t) u = 0.294, {1) k s = 1. for £ = -1, 0, 1. (2.178) Along the free edge at the tip of the cantilever plate, we assign a uniformly distributed resultant couple (-i)m* 12 = 189.8, which corresponds to the theoretical value of the tip moment to bend a beam equivalent to the thinner part L a of the plate into a full circle. Figure 2.10. Two-layer plate with ply drop-off, Aspect ratio A = 20 : 1 : (1, 0.5): Geome- try and assigned couple. As mentioned above, since the connection between the thickness part and the thin- ner part at the ply drop-off of the plate is flexible, the assigned resultant couple is higher than what is needed to roll the thinner part of the plate into a full circle. Thus it is ex- pected that the tip of the plate will be rolled past the ply drop-off location. The computed displacements of the nodes at the ply drop-off and at the tip of the plate are reported in Table 2.4. In the case where a full circle is obtained, point B in Figure 2.10 should come back to coincide with point A (in a projection onto the plane (f 1 , £ 3 )); in such case, the displace- ment of point B should be u\B) = -10 - v}(A) = -10 - 0.969 = -10.969, where the value of u 1 (A) = -0.969 comes from Table 2.4. The computed displacements u 1 for point B is, however, equal to (-10.789), thus corresponds to a relative error of 1.6% compared 56 Table 2.4. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5). Displace- ments at the ply drop-off (point A in Figure 2.10) and at the tip (point B in Figure 2.10). Node e £ 2 e u 1 u 2 u 3 101 10. 0. 0. -0.9689 7.875 x 10~ 3 -3.705 105 10. 1. 0. -0.9689 -7.875 x 10- 3 -3.705 201 20. 0. 0. -10.789 8.768 x 10- 3 -3.937 205 20. 1. 0. -10.789 -8.768 x 10" 3 -3.937 Figure 2.11. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll down maneuver . Isometric view of deformed shapes (Peeling of a banana). to the value of (—10.969) mentioned above. The deformed shape of the plate, shown in Figure 2.1 1, evokes the action of peeling a banana. To display the effects of anticlastic curvature that the sandwich shell elements can capture, we refine the discretization to 160 sandwich shell elements. The three-dimensional rendering of the deformed shapes are given in Figures 2.13, Figures 2.14, and Figures 2.15. The anticlastic curvature can be seen clearly in Figure 2.13 and Figure 2.14. Since the top surface of the plate is stretched in the f 1 direction, when the plate is roll down, by the effect of the Poisson's ratio, this top surface experiences a contraction in the £ 2 direction. The reverse is for the bottom surface of the plate (i.e., a contraction in the direction and a stretching (expansion) in the £ 2 direction). The combined effect of stretching and 57 Figure 2.12. Ply drop-off problem. Cantilever sandwich shell with drop-off subjected to tip moment: Peeling of a banana. contracting of the top and bottom surfaces of the plate in the £ 2 direction is the result of bending in the f 1 direction. To quantify the anticlastic curvature, and to compare the result with a calculation employing 3-D solid elements using the nonlinear finite element code ABAQUS, we look at the difference in the transverse displacement u z of two points located at f 1 = 13.5 (see Figure 2.15), one point at £ 2 = (i.e., at the outer lateral edge of the plate), and the other point at £ 2 = 0.5 (i.e., in the middle of the plate in £ 2 direction): u 3 (13.5, 0.5) - u 3 (13.5, 0) = -6.68096 - (-6.70291) = 0.022 . (2.179) The quantity in (2. 1 79) is to be compared to the quantity in (2. 1 80) obtained from ABAQUS. We note that the resultant couple needed to roll the thin part of the plate into a full circle is (-i)m* 12 « 177.5 which is 94% of the magnitude of the tip moment needed to roll an equivalent clamped beam into a full circle. This lower magnitude is due to the flexibility by the plate at the ply drop-off line, as discussed earlier. To compare the results obtained with our sandwich shell element, we solve the same problem using the solid elements in the nonlinear finite element code ABAQUS. In our 58 z It— >a Figure 2.14. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll down maneuver. 3-D rendering of the deformed configuration. Anticlastic curvature viewed from f 1 direction. ABAQUS model, we employ 960 C3D8I (8-nodes) linear brick elements, with 1453 nodes. These elements belong to the class of incompatible mode formulation. The final value of the resultant couple assigned along the force edge at the tip of the plate is 189.8 59 = 13.5 Figure 2.15. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll down maneuver. 3-D rendering of the deformed configuration. Anticlastic curvature viewed from £ 2 direction. (See Figure 2.19), which corresponds to that obtained from beam theory. Figure 2.16-2.18 provide various views of the final deformed configuration of the ABAQUS model. Figure 2.16. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll down maneuver. ABAQUS model using 960 solid elements with incompatible modes. Undeformed and deformed configuration. Observer's viewpoint: (1,-1,1). To quantify the anticlastic curvature, and to compare this quantification to the re- sult obtained using the sandwich shell elements, we again consider the nodes having the coordinates f 1 = 13.5, £ 3 = 0, which lie on the top surface of the thinner part of the 60 Figure 2.17. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll down maneuver. ABAQUS model deformed configuration. Observer's viewpoint: (1, 0,0). Figure 2.18. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll down maneuver. ABAQUS model. Undeformed and deformed configuration. Observer's viewpoint: (0,-1,0). two-layer plate. In the deformed configuration, these nodes are close to the points having the lowest spatial coordinate x 3 (or the z coordinate in Figure 2.15 and Figure 2.18). The displacements of these nodes are given in Table 2.5. Parallel to (2. 179) for sandwich shell elements, the anticlastic curvature in the ABAQUS model can be quantified using the results in Table 2.5 as follows u 3 (13.5, 0.5) - u 3 (13.5, 0) = -6.5948 - (-6.6159) = 0.022. (2.180) The above result agrees well with that obtained from the sandwich shell element. 61 Deformed plate M = F x d Figure 2 19 Two-layer plate with ply drop-off. Aspect ratio A » 20 : 1 : (1,0.5): ABAQUS solid model. Assigned forces at plate tip to create a resultant couple in the roll-down maneuver. . Table 2 5. Two-layer plate with ply drop-off. Aspect ratio A -20: 1: (1, 0.5): Anticlastic curvature from ABAQUS solid model. Nodal displacements of nodes having coordinates f 1 = 13.5, £ 3 = 0. e e u 1 u 3 13.5 0.00 -6.007 -6.780 13.5 0.25 -6.007 -6.764 13.5 0.50 -6.007 -6.758 13.5 0.75 -6.007 -6.674 13.5 1.00 -6.007 -6.780 For the thinner part of the two-layer plate to roll into a complete circle, point B in Figure 2.10 must roll back to coincide with point A which is itself moved by the deformed plate. To compare the results obtained using sandwich elements and those obtained using an ABAQUS model, we gather the coordinates of point A and B in the final deformed configuration, corresponding to the resultant couple of M = 189.8 in Table 2.6 below. 2.4.2.4. Tw o-layer plate with ply drop-off: aspect ratio A = 20 : 10 : (1, 0-5) We now consider the same plate as in the previous section, but with a width ten times larger (i.e., W = 10.), instead of W = 1. as in the previous section. All other geometric dimensions and material properties remain identical to those in the previous 62 Table 2.6. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1, 0.5). Compari- son between sandwich elements and ABAQUS solid model. Distance between point A and point B in the final deformed configuration. £ A = (10, 0, 0) , £ B = (20, 0, 0) Sandwich elements 9.03 0.00788 -3.71 ABAQUS model 8.64 0.0167 -3.56 <F(€a) * 3 (£b) Sandwich elements 9.21 0.00877 -3.94 ABAQUS results 8.61 0.00930 -3.53 Distance between A and B = * ( £ /i) ~ * ( £ b)| Sandwich elements 0.293 ABAQUS results 0.0459 section, see Eqs. (2.175), (2.176), (2.177), and (2.178). The distributed couple ( -i)m* 12 assigned to the free edge of the plate tip is set as before to (-i)m* 12 = 189.8, which is the resultant couple that will roll an equivalent beam into a full circle. To discretize the two-layer plate, we now employ 200 sandwich shell elements: 100 elements before the ply drop-off, and 100 elements after the ply drop-off. The deformed shapes of the plate are shown in Figure 2.20- 2.23. Figure 2.20 depicts the deformed shape in an isometric view. One can clearly see the anticlastic curvature in the £ 2 direction in this figure, as well as in Figure 2.22, which is the projection of the deformed shape on the (£ 2 , £ 3 ) plane. This anticlastic curvature is the effect of the Poisson's ratio. The top surface of the undeformed plate is extended in the £* direction, this extension induces a contraction in the transverse £ 2 direction, and thus the downward curvature is clearly seen at the bottom of the deformed plate in Figure 2.22. Opposite to what take place at the top surface, the compression of the bottom surface of the plate in the £ x direction induces an extension in the £ 2 direction, thus resulting in the lateral bulging of the plate, as seen in Figure 2.22. To quantify the anticlastic curvature similar to (2.179) and (2.180), we consider the transverse displacement u 3 of the nodes at £ r = 14 and £ 3 = 0, which lie near the bottom z III ,x Figure 2.20. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5): Roll- down maneuver. Sandwich shell elements 3-D rendering of deformed shape. Isometric view. B A Figure 2.21. Two-layer plate with ply drop-off. Aspect ratio A — 20 : 10 : (1,0.5): Roll-down maneuver. Zoom-in on the ply drop-off point. of the deformed configuration of the two-layer plate (see Figure 2.23) u 3 (U, 5) - u 3 (14, 0) = -6.62581 - (-6.64764) = 0.02183. (2.181) z Figure 2.22. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5): Roll-down maneuver. Sandwich shell elements. 3-D rendering of deformed configuration. Projection down the f 1 axis. To clearly depict the deformation at the ply drop-off and the tip of the deformed plate around the area of the ply drop-off, we provide a zoom-in figure on this area in Figure 2.21. The distance between point A and point B will be used to compare the results obtained with sandwich shell elements and those obtained with an ABAQUS solid model, which is composed of 2400 incompatible (solid) linear brick elements (type C3D8I). The final moment at the plate tip assigned to the ABAQUS solid model has a magnitude of 1898, which is obtained for the roll-down of an equivalent beam. Various views of the deformed configuration obtained with the ABAQUS model are depicted in Figure 2.24, Figure 2.25, Figure 2.26. The distance between point A and point B in the deformed configuration as obtained both from sandwich shell elements and from the ABAQUS model are given in Table 2.7. From the above examples, we found that the present sandwich shell formulation gives good results on displacements with very coarse mesh, when compared to the 3-D converged results from ABAQUS. For the interlaminar stress analysis, due to the kinematic assump- 65 Figure 2.23. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5): Roll-down maneuver. Sandwich shell elements. 3-D rendering of deformed configuration. Projection down the £ 2 axis. Figure 2.24. Two-layer plate with ply drop-off. Aspect ratio A — 20 : 10 : (1,0.5): Roll-down maneuver. ABAQUS solid model. Undeformed and deformed configurations. Isometric viewpoint : (-4,-7,-3) tion, the present formulation is expected to be more accurate than the single-layer shell model, especially for thick and moderate thick shells. 66 Figure 2.25. Two-layer plate with ply drop-off. Aspect ratio A - 20 : 10 : (1,0.5): Roll-down maneuver. ABAQUS solid model. Projection down the f 1 axis of deformed configuration. Figure 2.26. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5): Roll- down maneuver. ABAQUS solid model. Projection of the deformed configuration along the £ 2 axis. 67 Table 2.7. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5). Compari- son between sandwich elements and ABAQUS solid model. Distance between point A and point B in the final deformed configuration. £ A = (10, 0, 0) , £ b = (20, 0, 0) $ 3 Ua) Sandwich elements 9.16 0.0112 -3.5 ABAQUS model 9.16 0.0179 -3.53 Sandwich elements 9.07 -0.0345 -3.41 ABAQUS results 8.23 -0.0294 -2.83 Distance between A and B = $ ( f a) ~ * ( £ b)\\ Sandwich elements 0.142 ABAQUS results 1.17 CHAPTER 3 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER COMPOSITES : STATICS 3.1. Introduction The analysis of general shell structures have been of interest for several decades. There is a continuing need to develop more reliable, accurate and efficient shell element, especially for analyses of composite structures covering a wide range of physical scales (in- cluding MEMS 5 ) and material and geometric nonlinearities. Structures made of laminated composites continue to be of great interest for engineering applications. For accurate anal- yses of composites with a large number of layers, industry routinely employs FE meshes with one solid element per ply in the thickness direction, and with element aspect ratio less than 10 (Figure 3.1). It is therefore highly desirable to develop efficient finite elements that are accurate at extreme aspect ratio to significantly reduce the computational effort. Figure 3.1. Composite structure with 500 plies in the thickness direction; the ply thickness is around 10 -3 zn. Shell element formulations have been mainly developed within the context of the so-called degenerated shell concept and the classical shell theory (Buechter and Ramm [1992]). Both formulations are based on the common kinematic assumptions of inextensi- 5 MEMS stands for MicroElectroMechanical Systems. 68 69 bility in the thickness direction and the zero transverse normal-stress condition. 6 Although these approximations led to very good results in most cases, several difficulties could arise: (i) Complex 3-D material models: the zero transverse normal stress condition must be im- posed, (ii) Boundary conditions and finite rotations: use of rotational degrees of freedom (dofs); especially those normal to the boundary, to describe soft support and hard support (e.g., Zienkiewicz and Taylor [1991, p.92]); complex update algorithms for finite rota- tions in geometrically-exact stress-resultant formulation (e.g., Vu-Quoc and Deng [1995], Vu-Quoc and Ebcioglu [1996], Vu-Quoc and Ebcioglu [2000a], Vu-Quoc and Ebcioglu [20006], Vu-Quoc, Deng and Tan [2000]). (iii) Transverse normal stress: inconsistently a posterior computation based on the computed in-plane stresses (see Reddy [1997, p.345], and e.g. in the localized effects due to the concentrated surface loading and the delam- ination of composite shells), (iv) Combination with regular solid elements: Transition elements are needed to connect rotational dofs and displacement dofs (e.g., Kim, Varadan and Varadan [1997] and the contact problem), (v) Through-the-thickness stress distribu- tion in laminated composite with dissimilar materials: poor accuracy because of straight director assumption (Bischoff and Ramm [2000]). Accurate and robust low-order shell elements have always been in high demand for development and for use in engineering analysis (e.g., DYNA3D [1993], NIKE3D [1995]), particularly when complex nonlinear 3-D constitutive relations can be incorporated with- out the added requirement to satisfy the constraint of zero transverse normal stress. Three possible shell kinematic descriptions have been proposed: (i) The displacement of the refer- ence surface together with the extensible transverse director (Simo, Rifai and Fox [1992], Betsch, Gruttmann and Stein [1996]). (ii) The displacement of the reference surface to- gether with the displacement vector of the tip of a director (Braun, Bischoff and Ramm [1994], Roehl and Ramm [1996], Bischoff and Ramm [1997], Bischoff and Ramm [2000]). (iii) The displacement of the top and bottom surface of the shell (e.g. Hauptmann and 6 Stress-resultant shell formulation can be generalized to account for thickness change, which relaxes the zero transverse normal stress condition (e.g., Simo, Rifai and Fox [1992]). 70 Schweizerhof [1998], Klinkel, Gruttmann and Wagner [1999], Ramm [2000]). The kine- matic descriptions (ii) and (iii) are attractive since they avoid the complex rotation updates in stress-resultant elements. On the other hand, the kinematic description (iii) provides a natural way to connect to regular solid elements without the need for transition elements; such feature can benefit the detailed modeling of shells with patches of piezoelectric or viscoelastic materials. The present solid-shell element has the same displacement dofs as in regular lin- ear (8-node) brick solid element. Displacement-based solid elements are known to have poor performance in bending-dominated situation, such as in thin shells. To obtain the same performance as stress-resultant shell formulations with plane stress assumption (e.g., Vu-Quoc, Deng and Tan [2000]), the Enhanced Assumed Strain (EAS) method and the Assumed Natural Strain (ANS) method are employed here. To improve the bending behavior of low-order elements, the EAS method based on the Fraeijs de Veubeke-Hu-Washizu functional was proposed by Simo and Rifai [1990]. For large deformation analyses, there are two ways to introduce the EAS method: (i) enhancing the deformation gradient F (Simo and Armero [1992], Miehe [1998b]), and (ii) enhancing the Green-Lagrangian strain tensor E (Bischoff and Ramm [1997], Klinkel and Wagner [1997], Klinkel et al. [1999] etc.). From the computational standpoint, the latter is simpler and more efficient, even though our numerical experience indicates that both approaches lead to the same numerical results when the same EAS parameters are used. 7 To incorporate 3-D constitutive laws in shell formulations, the transverse normal strain must have at least a linear distribution over the shell thickness; otherwise, the so-called Poisson-thickness locking would occur (Zienkiewicz and Taylor [1991, p. 161], Bischoff and Ramm [1997]). To relieve the Poisson-thickness locking, two methods were proposed in recent years: (i) Assuming a quadratically distributed displacement field over the shell thickness (Parisch [1995]), which then introduced an additional kinematic parameter, and (ii) using the EAS 7 Noted that the EAS method based on the displacement gradient as proposed in Miehe [19986] and used in Miehe and Schroeder [2001] does not pass the bending patch test, and there is no easy way to remedy this problem, see the next paragraph for the details. 71 method to enhance the transverse normal strain (Buchter, Ramm and Roehl [1994]). In our formulation, we enhance the transverse normal strain by the HAS method to include bilinear terms ^ 3 and £ 2 £ 3 in terms of material coordinates. To improve the membrane bending behavior, we also enhance the membrane strains in the similar manner as in Simo and Rifai [1990]. On the other hand, to make the formulation more efficient, we propose a modified HAS method while still keeping the same level of accuracy, where the inverse of element Jacobian matrix and the Jacobian at the element center are no longer necessary. Further- more, the present eight-node solid-shell element relies on a new optimal seven-parameter EAS-expansion (for the transverse normal strain and for the membrane strains) together with an ANS method (for the transverse shear strains); the present formulation is shown to pass both the membrane patch test and the out-of-plane bending patch test. It should be noted, however, that while the 30-parameter HAS expansion of Klinkel and Wagner [1997], the five-parameter HAS expansion of Miehe [1998*] and of Klinkel et al. [1999] pass the membrane patch test, all of them fail to pass the important out-of-plane bending patch test. For the HAS approach using enhancing deformation gradient, we develop an HAS expansion by superposing the enhancing converted basis to the compatible converted basis, and then present a formulation that is much simpler than that employed in Miehe [1998*] (see Section 4.3). Two ANS modifications on the compatible covariant strains are employed to elimi- nate the locking effects from the compatible low-order interpolations. ANS interpolation is the most successful tool to overcome the shear-locking effect in the 4-node displacement- based shell elements, even for initially distorted meshes (MacNeal [1978], Hughes and Tezduyar [1981], Dvorkin and Bathe [1984], Parisch [1995]). We apply an ANS inter- polation of the compatible transverse shear strains to treat shear locking. In the case of curved structure with geometric nonlinearity, there is another locking effect: The so-called curvature-thickness locking (Bischoff and Ramm [1997]), which is also known as the trape- zoidal locking (Sze and Yao [2000]); this type of locking can be avoided by introducing the 72 ANS interpolation of the compatible transverse normal strain, as proposed by Betsch and Stein [1995]. Such treatment can improve the performance of the formulation in Parisch [1995] and Hauptmann and Schweizerhof [1998]. The features of this solid-shell formulation are summarized as below: . The kinematic description involves only displacement dofs that require no complex finite rotation update and no transition elements to connect solid-shell elements to regular solid elements (Hauptmann and Schweizerhof [1998]). . The use of covariant Green-Lagrange strain tensor, without neglecting any higher order terms (e.g., as in Bischoff and Ramm [1997]). The stress and strain terms quadratic in £ 3 become important in the analysis of relatively thick shells, for strong curvatures or in the presence of large strains together with bending deformations (Buchter et al. [1994]). . All stress and strain components are accounted for, thus allowing for an implementa- tion of unmodified 3-D nonlinear constitutive laws, without the need for applying the plane-stress constrain. The strain-driven character of the formulation also makes it easier to implement nonlinear constitutive models, as compared to the hybrid finite- element formulations (Simo, Kennedy and Taylor [1989]). . In contrast to EAS formulation based on the deformation gradient F (see Section 4.3), EAS formulation based on enhancing the Green-Lagrange strain tensor (together with the use of the second Piola-Kirchhoff stress tensor) are much simpler (Simo and Armero [1992] and Andelfinger and Ramm [1993]). . An ANS method applied on the transverse shear strains is used to relieve the trans- verse shear-locking problem (Dvorkin and Bathe [1984]), whereas an ANS method applied on the normal strain components is used to remedy the curvature thickness locking problem (Betsch and Stein [1995]). 73 In addition to the above features, our new contributions to the field are specifically listed as below: • Optimal (minimum) number of EAS parameter to pass the patch tests for both the membrane response, and the out-of-plane bending: (i) three EAS parameters on the transverse normal strain to remedy the Poisson-thickness locking, and (ii) four EAS parameters on the membrane strains to remedy the in-plane bending behavior. • Efficient EAS method that avoids the computation of the Jacobian at the element center, and that no inverse of the Jacobian matrix at the element center is needed. • By using the tensor form, we prove the equivalence of the 2-D plane elasticity el- ements of Simo and Rifai [1990], of Taylor, Beresford and Wilson [1976], and our new enhancing formulation. • We justify through numerical experiments the relative importance of the separate use of the EAS method and the ANS method, as compared to the pure displacement formulation, and more importantly, the combined use of both the EAS method and the ANS method in obtaining accurate results for plate bending problem over a large range of aspect ratios. The comparison among the above various solid-shell formulations is listed in Table 3.1. 8 Table 3.1. Comparison of various solid-shell concepts. Bending Patch test Loc king-free Absence of rot. dofs/ Disp. dofs only Higher-order terms in thickness coord. Absence of pre- integration Model parameter- space dimension Optimal EAS Present clement yes yes yes yes yes 3-D yes RammcL a].[1997) yes yes yes no no 2-D no Schweizcrhof el. al.|1998] yes no yes yes yes 3-D no BelscheL al.|!996] yes yes no yes yes 2-D no Miehe |1998] no yes yes yes no 3-D no Parameter-space dimension is defined as: 1-D (beam), 2-D (stress-resultant plates and shells) and 3-D (solids); Deformation-space dimension is defined as: 2-D (planar deformation), 3-D (general deformation) 74 The outline of the present chapter is as follows. After a presentation of the kinematics assumption and the formulation based on the Fraeijs de Veubeke-Hu-Washizu (FHW) vari- ational principle (Felippa [2000]) in Section 3.2, we discuss the finite-element discretiza- tion and its implementation in Section 3.3. A review of the EAS method together with our proposed modification is presented in Section 3.4. We present the numerical results in Section 3.5. 3.2. Kinematic Assumption and FHW Variational Formulation The extension of the EAS method to geometrically non-linear problems by Simo and Armero [1992] employed an enhancement of the deformation gradient F and thus a multi- plicative decomposition An alternative line of formulation for geometric and material non- linearities based on the enhancement of the Green-Lagrange strain E leads to particularly efficient computational effort (see, e.g. Bischoff and Ramm [1997]), with practically the same results. We describe below the kinematics of a solid shell in curvilinear coordinates and review the three field FHW variational principle and its role in the EAS method. 3.2.1. Kinematics of Solid-Shell in Curvilinear Coordinates To overcome the known problems associated with the rotational degrees of freedom in traditional shell elements, the shell kinematics of deformation is described by using the position vectors of a pair of material points at the top and at the bottom of the shell sur- face. In this kinematic description, a straight transverse fiber before deformation remains straight after deformation. Such transverse fiber between two corresponding nodes at top and bottom surfaces needs not be normal to the shell mid-surface before deformation, as well as after deformation. We distinguish here three configurations of the shell: (i) the material configuration, which is the biunit cube, (ii) the initial (or undeformed) configura- tion, which could be curved, and (iii) the current (or deformed) configuration (see Fig.5 in Vu-Quoc and Ebcioglu [20006]). The initial (undeformed) three-dimensional continuum of the shell geometry (Figure 3.2) is described by 75 □:«H,l]xH,l}xH,U, (3.1) where X (•) is the mapping from the biunit cube □, parameterized by the material coordi- nation (f 1 , £ 2 , £ 3 ), to the initial configuration. The image X (£) of a point £ = (f 1 , £ 2 , £ 3 ) 6 □ is next represented by a linear combination of the position vectors X u (•, •) and Xi (•, •) of a point in the upper material surface £ 3 = + 1, and a point in the lower material surface Figure 3.2. Initial (undeformed) configuration of solid shell: convective coordinates f and position vectors X u and X/. The kinematics of the present formulation is the same as in (Hauptmann and Schweiz- erhof [1998]), and related to the director formulation (e.g. Bischoff and Ramm [1997]), if one rewrites (3.1) as follows xw = I [x.{e,e)+x, (e,e)} +le[x.{t 1 ,?)-x, (?,?)} where X m is the position vector of the mid-surface in the initial configuration, D the unit director, and h (£*, £ 2 ) the shell thickness. 76 Similarly, in the current (deformed) configuration, the geometry of the solid shell is described by 1 2 1 2 ;«.(^)+*i(^,c a )]+-5? m*\ (^,e 2 ); ;(i+«*)«-(€ 4 .^)+(i-^)*»(^) (3.3) where ar(-) is the mapping from the biunit cube parameterized by (£\£ 2 ,£ 3 ) (material configuration), to the current (deformed) configuration, x u (•, •) andxj (•, •) the position vectors of the deformed upper and lower surface of the solid shell, respectively. The initial configuration is related to the deformed configuration (Figure 3.3) by the displacement field u as follows x(t) = X(£) + u($). (3.4) The convected basis vectors G h i = 1, 2, 3, in the initial configuration B are related to the position vector X and the converted coordinates <f by dx(0 and satisfy the following relations 2 = 1,2,3, (3.5) Gi'G j = S{ , Gi • Gj = dj , i,j = 1,2,3, (3.6) where Gy are the components of the metric tensor with respect to the basis G { <g> G j in the initial configuration B . configuration. To simplify the presentation, we will omit the argument (£) and simply write G { and G\ The covectors G i can be obtained by the following relation G l = G j Gj , with Qij = [Gij]- 1 € (3.7) similarly, the convected basis vectors gi in the current configuration B t are obtained by using (3.4) and (3.5) as follows dx du (3.8) 77 B Figure 3.3. Solid-shell: material configuration B, initial configuration Bo, and deformed configuration B t . and satisfy the following relations 9 % -9 1 = % , g.-Qj = 9ij , i, j = 1, 2, 3 , (3.9) where c/jj are the components of the metric tensor with respect to the basis g { <g> g j in the current configuration B t . The covectors g i can be obtained as follows g l = g l3 gj , with [<^j = [^-p e -1 ^ hd3x3 (3.10) The deformation gradient expressed in converted basis vectors g { and takes the form (3.11) Using (3.1 1), (3.6), and (3.9), we can then write the (compatible) Green-Lagrange strain 78 tensor with respect to the convective coordinates as follows E c = l - [F T F - I 2 ) = i [(G* ® g) ( 9j G j ) - CjjG' ® G j = \ im ~ G l ®G j = E^G { <g> G J , (3.12) where is the covariant components of the strain tensor E c , and JT 2 the identity tensor expressed with respect to the convected basis G l ® G-* using the relations e p — Gf G ! and e 9 = GjG- 7 as follows ff 2 = <$ P<? e p ® e 9 - ^G^GjG* ® G J = GJgJG* ® G J = GjjG' G J , (3.13) Using (3.8), the metric-tensor components in (3.9) 2 can be expressed in terms of the convected basis vector Gj and the displacement vector u as follows _ _ du du _ <9w <9u Substituting (3.14) into (3.12), the covariant components 25& of the compatible Green- Lagrange strain tensor E c read as The second Piola-Kirchhoff stress tensor S is conjugated with the Green-Lagrange Strain E c , and can be expressed with respect to the basis G { <g> Gj as follows S = S^G l ®G j , (3.16) with 5 U being its contravariant components. 3.2.2. Variational Formulation of EAS Method In this section, we provide a brief overview of the EAS method, which has the three- field Fraeijs de Veubeke-Hu-Washizu (FHW) 9 variational principle with the following 9 We refer the readers to Felippa [2000] for a history on the contribution of Fraeijs de Veubeke to the formulation of multifield variational principles. 79 functional as the point of departure «• (b* - u) pdV + / (u* - u) -idS - / wt*dS (3.17) where the displacement it, the Green-Lagrange strain and the second Piola-Kirchhoff stress S are the independent variables. In (3.17), the expression of E c (u) in terms of u is given in (3.12) and (3.15); W s is the stored strain energy, t the traction vector, u* and t* the prescribed displacement on the boundary S u and the prescribed traction on the boundary S a , respectively. All variables here are expressed in the initial configuration B , with S a l) S u = dB (i.e, the closure of the union of the boundary S u and the boundary S a forms the complete boundary of B ). The next step in the EAS method is to introduce an enhancement (or enrichment) to the Green-Lagrange Strain E° and, as a result, of the variation SE C as follows 8E = 6E C + SE , where E and SE represent the enhanced strain and its variation, whereas E and SE sent the enhancing strain and its variation. Introduction of (3.18) into (3.17) yields E = E C + E , (3.18) (3.20) where it is noted that the traction t is actually a function of 5(see also Malvern [1969, p.69]), and not an independent variable. Consider a perturbation of the displacement u as follows: u e (X, t) = u (X, t) + eSu (X) , (3.21) since the variation Su is only a function of space, and not of time, the variation of the acceleration u is then Su = , (3.22) therefore the variation of U in (3.20) with respect to the displacement u is d_ de u(u S: E,s) = /^f: (^^-'5u\dV- JSu'(b*-u)pdV - J SwtdS - J 5wt*dS , ySu, (3.23) dW s Since the second-order tensor — — - is symmetric, the first term in (3.23) can now be rewrit- uE ten as follows = ^f"^")-*^)*' (3 - 24) where the first equality is shown in Remark 3.1, and the second equality is obtained with Leibniz rule. Remark 3.1. With the deformation gradient F (u) expressed with respect to the basis ej (8) Ej (see Figure 3.3), we obtain the following relations F m F'ja ® E J , F T = F)E J ® et , (3.25) F T F = FigijF'E 1 <g> E J , with g ij = e l -e j , (3.26) dF dF* SF = —8u = -^-ei®E J 8u. (3.27) Note that, if {ei, e 2 , e 3 } were a system of orthonormal vectors, we would have g l i = 6ij, however, in general, it is not necessary to have {ej, e 2 , e 3 } orthonormal. In this remark, we retain the notation cfi for generality. We define a symmetric second-order tensor S as 5 := Je = sABEa ® Eb ' (128) By using (3.28), (3.12), and (3.26), we can then rewrite the first term in (3.23) as follows Since S AB = S BA from the symmetry of S, (3.29) becomes ^:6E< = F\S AB giJ ^6u, (3.30) With the following component form of FS using (3.25)i and (3.28), that is, we obtain FS = F l jS JB ei®E B , (3.31) and then the following expression BF f) FS : 5F = FSl —Su = F l A S AB gij ^8u . (3.32) ou J du Comparing (3.32) to (3.30) and using SF = GRAD (Su), we arrive at dW ~ : SE C = FS : SF = FSl GRAD (6u) . (3.33) oh/ I Using the divergence theorem , the integration of the first term in (3.24) becomes / Div { f ^e} Su ) dV= I { F ^ Su ) ' ndS ' (334) B ^ ' dB * ' where n is the outward normal vector to the surface 8B . 82 With (3.24) and (3.34), we can write the variation of II in (3.23) with respect to u as |n(«.,S,s)| - / n ( F ^\. SudS -Jn 1 J F m.su < tv - J Su- (b* - u) pdV - J Su-tdS - J 8wt*dS =/ -i Bo n* | Div mM8+ J [-('?*)-* Sa ' 'SudS + pb* — p'u '5udV = 0, V5u, (3.35) thus the Euler-Lagrange equations resulting from the variation of IT with respect to u in above (3.35) are ( dW s \ Div + p6* = pit in £> , (3.36) (3.37) (3.38) Then make the variation of IT of (3.20) with respect to E by the perturbation it follows that E £ = E + eS E , (3.39) l n =/( ' aw. dE - s : , v<5£ (3.40) the Euler-Lagrange equation associated with (3.40) is dE ' (3.41) Taking the variation of II of (3.20) with respect to S by the perturbation S £ = S + e6S , (3.42) 83 we obtain II (uE, S e ) m / 6S : EdV + j (u*- tt) -StdS , V6S , (3.43) 6=0 Bo S u where it is noted that t depends on S, then the Euler-Lagrange equations associated with (3.43) are E = 0, (3.44) u = u* on S u , (3.45) where in general E when the finite element approximation is introduced. 3.3. Finite-Element Discretization In this section, we present the weak form and finite-element approximation of the proposed solid-shell element. The orthotropic constitutive law of laminated composite is then derived in convective coordinates. To avoid shear locking, we use the assumed natural strain (ANS) method by Dvorkin and Bathe [1984] for the transverse shear strains. To remedy the curvature-thickness locking (Bischoff and Ramm [1997]), we adopt the ANS method by Betsch and Stein [1995] for the transverse normal strain. 3.3. 1 . The Weak Form of Modified Two-Field FHW Functional By designing the approximation for the stress field S and the approximation for the enhancing strain E c such that the following orthogonality condition holds: / 5 : EdV = . (3.46) Bo then the number of independent variables in the functional II in (3.20) is reduced to two, that is n (u, e) = jw s (E c («) + E) dV Bo - y u- (6* -u)pdV + J (u* -u)-t(S)dS- J u-t*dS , (3.47) Bo 5 U S, 84 leading to the following total variation SU («, E) = 6U mass (u) + 8U stlff (u, E) + SU ext (u) = , (3.48) where from (3.35) and (3.40), we have SU mass = Jdu-updV, (3.49) Bo 5n stiff (u, E) = j (SE C (u) + 5E) : -^W s (E c (u) + E)dV , (3.50) Bo SU-txt = - j 6u-b*pdV - J 5wt*dS . (3.51) 3.3.2. Spatial Discretization Let the initial configuration B be discretized into nonoverlapping nel elements B^ nel ,s with numnp nodes, such that B « |J Ba . Let /i denotes the characteristic size of the e=l finite element discretization. In the element domain Bq, the displacement u, its variation Su, and increment Aw are interpolated as follows u^u h = N(Z)S e \ (3.52) 8u ^Su h = N (£) W (e) , Au w Aw' 1 = AT (£) Ad (e) , (3.53) where AT is a matrix containing the basis functions restricted to element Bq \ and S e) € R 3x numnp a matr j x containing the nodal displacements. The readers are referred to the Appendix A.l for the details. The velocities u and accelerations u are also interpolated by using the same shape functions and the corresponding nodal values that is h « & = N (0 d (e) , u^u h = N(i%) d (e) . (3.54) In what follows, for simplicity, we will omit the superscript h in u h , and simply write u. Within a typical element (e), the variation and the increment of the (compatible) Green-Lagrange Strain E c is related to the variation and the increment of displacement, respectively, based on (3.15), as follows fa&Li = B KO 6di£) - { A ^} 6xl - B ( die) ) ^ , (3.55) where the components of had been arranged into a 6 x 1 column matrix according to the Voigt ordering (Brillouin [1946, p.221]) } = {E c n , E c 22 , 2E C 12 , E c 33 , 2E C 23 , 2E C 13 } T , (3.56) and where B is the deformation-dependent displacement-to-strain operator, where detailed expression is given in the Appendix A. 1 . We denote the admissible variation of the element EAS parameter column matrix ojW g r»mw associated with the enhancing strain by 6a&, where neas is the num- ber of EAS parameters a (e) . Interelement continuity is not required for the enhancing strain E, where components can be approximated via an enhancing strain interpolation matrix Q and the element EAS parameters a (e) ; an interpolation applies to the variation and the increment of E, that is {4} 6X1 = <5 (0 " (e) , {SE tJ } 6xi = 9 (0 Sa^ , {A4) 6xi = g $ Aa<< (3.57) The number of internal parameters ot^ and the interpolation matrix g will be discussed in Section 3.4. 3.3.3. Linearization of the Discrete Weak Form The consistent tangent operator in the Newton solution procedure is constructed by taking the directional derivative of the weak form at a configuration {k) u in the direction of the increment A {k) u, where the left subscript k designates the iteration number. The tangent operator can be viewed as the summation of the material and geometric tangent operators. The geometric part results from taking the variation of the geometry while hold- ing the material constant, whereas the material part results from taking the variation of the material while holding the geometry constant. 86 Applying a standard finite-element procedure to discretize the weak form (3.48), we obtain the following expression at the element level for the static case (u = 0) 6U^=SU%+8U^ t = 0, (3.58) with the stiffness part SU^ lff from (3.50) and the external-force part SU { ^ t from (3.51) written as follows 5U% = J 6{E^ T {s^}dV+ I 5 {E 13 } T {S^} dV , (3.59) SUii = - J 5u-b*pdV- J Su-fdS, (3.60) where we used the symbol definition, which corresponds to the second Piola-Kirchhoff stress, s ■= a£> (3 - 61) to alleviate the notation, and where the column matrix {S ij } has its coefficients arranged in the same Voigt ordering as in (3.56) {S«} m [S n ,S 22 ,S 12 ,S 33 ,S 2 \S 13 ] T . (3.62) It should be noted that the symbol S in (3.61) is simply used in the place of - , and is dE not the independent variable in the Euler-Lagrange equation (3.41). Remark 3.2. The linearization of the weak form 8U. (u, E^j can be accomplished by the truncated Taylor series about the kth iterate ( , ( k )E ) : <5n( (fc+1 )u, (k+i)E) « sn(( k )u, (k) E) d(8U) • (Au, AE) (u= (k) u,E= w E^j d (it, E) = <5n( (A) u, [k] E) -f Z>(<ffi)( (fc)U , {k) E) • [Au,AE) , (3.63) 87 where Au = {k+1) u - {k) u, AE = {k+l) E - {k) E . To compute the increment (Aw, AE) in the Newton's solution process, we simply set the expression in (3.63) to zero. To alleviate the notation, we will omit the left subscript k designating the iterative index. Using the approximation (3.52), (3.53), (3.55) and (3.57) in (3.63), the increments Ad [e) and Aa< e » can be computed in the Newton's solution process, as mentioned above, by using the following equation V(6IIU) (*>,««) • (AdM Aa<->) = ( Ad « Aa (e)) in which the variation SU% in (3.59) and SIlQ in (3.60) now take the form (3.65) (d« , tt W) with f% = J B T {s ii }dV, Ae) J EAS ~ aiS (<* (e) ) UUl J ext > with fi e x \ = f N T b*pdV+ I N T t*dS 8 :<<=> 8. .00 (3.67) (3.68) Thus the left hand side of (3.64) becomes V 1 K ' 8d (e) daW = K )T *2 + toW^fcW] , Ad (e) + [sd^kH + fa^SS] -AaW = [fcW AdW + fcW AaW] + feW [*W ArfW + *£> Aa M] , (3 69) Let the matrix of tangent elastic moduli C be defined as c = Qijkl dE kl \ j6x6 (3.70) 88 where C ljkl are the components of constitutive tensor C in the convected basis, and are subsequently arranged in the matrix C according to the ordering of the strain components in (3.56) and of the stress components in (3.62). Using (3.69), (3.66)i, (3.70), and (3.55), we obtain the following expressions df (e) *S = = / (G r S + B T CB) dV , (3.71) k ™ - - 1 BTcgdV • < 3 - 72 > s <«) where the matrix dB (S e) ) (which is a function of the coordinates £), and the stress matrix $ (which is related to the matrix {S* j } in (3.62)) have their detailed expressions given in the Appendix A.l. It is noted that the dimension of $ and G are 144 x 24 for the present element with six stress components, eight nodes per element, and three dofs per node. From (3.69), (3.66) 2 , (3.70), (3.73), and (3.55), we obtain the remaining parts of the stiffness matrix fc (e) °J EAS \jJe)] T f r TnnA\r 't***\ k au - (e) - [*^J = j y CBdV , (3.74) df (e) EAS = J Q T CgdV. (3.75) It follows from (3.64), (3.65), (3.67), and (3.69) that the discrete linearized system of equations to solve for the increments Ad (e) and Aa (e) is given by fcWAdW + fcWAaW = f%-f%, (3.76) fcWAifW + fcWAaW = (3.77) or in matrix form as ^ a I _ J / eit J stiff f(e) J EAS AaW = • (3-78) 89 Since the enhancing strain E is chosen to be discontinuous across the element bound- aries, it is possible to eliminate the EAS parameter increment Aa (e) at the element level, before proceeding to assemble the element matrices into global matrices. Solving for the increment Ao w using (3.77) Aa (e) - - [*£] ~* (f&s + k%&4#>) , (3.79) then substituting (3.79) into (3.76), we obtain the following condensed symmetric element stiffness matrix fc$? and the element residual force vector = $-{^J T !if' , *& (3-80) p (e) _ Ae) _ f (e) , [l.(e)l T ^(e)]- 1 f (e) nRn ' — J ext J stiff t [« QU J |« QQ J Teas- (3.9 1) An assembly of the element matrices fe^ and r (e) leads to the global system K T Ad = # , (3.82) nel nel with -KY = AfeF , # = Ar (e) , (3.83) where A denotes the finite-element assembly operator. The incremental displacement Ad can be solved by using (3.82), and the displace- ment d and d (e) updated. With (3.79), the incremental displacement Ad (e) is used to com- pute the increment Aa (e) , which is in turn used to update the EAS parameter a (e) . The details of this iterative procedure are provided in the Appendix A.l. 3.3.4. Material Law in Convected Basis For the Saint- Venant-Kirchhoff material, the fourth-order material tensor C is de- fined as the second derivative of the stored energy function W s with respect to the Green- Lagrange strain tensor E, r _ d 2 W s L - dEdE > ( 3 - 84 ) and the second Piola-Kirchhoff stress tensor S is then expressed as dW S=-^=C-.E. (3.85) 90 The constitutive relation of laminated composites can be described by using an or- thotropic material law. For that purpose, we express the components C ljfc ' of tangent elastic moduli tensor C relative to the fiber reference axis {ai, a 2 , a 3 } of a lamina, and arrange these components in a matrix fei (see, e.g., Reddy [1997, p.41] and Figure 3.4), using the same ordering of the strain components in column matrix form as in (3.56) (see also (3.62)). b 2 b 3 — a 3 Figure 3.4. A fiber-reinforced lamina and fiber reference axes {a^ a 2 , a 3 }. (Ollll £<1122 q £1133 £>1122 £2222 q £2233 C 1212 £1133 £2233 q £3333 £2323 o c 1313 6x6 (3.86) where the components C ljkl take the following expressions /Mill £l (1 ~ ^23^32) ^2222 _ E 2 (l- 1/13^31 ) /S3333 ^3 (1 ~ ^12^2l) " A ' C - A ' C = A ' £1122 #1 (^21 + ^31^23) AU33 _ E 3 Q 13 + U U U 2Z ) - 2233 _ E 2 (>32 + ^12^3l) A ".<'-<- A , O - , 91 A = 1 - V l2 V 2X - IA>3^32 - ^21^13 T 2^12^32^13 i /S1212 _ ^t2323 ^ /M313 /-> V — ^12 > ^ — <-*23 ) O — Lri3 , = ^ji^i , for (i, j — 1,2,3 , and i ^ j) , and Ei,E 2 ,E 3 are the Young's moduli in the principal material directions {ai,a 2 ,a 3 }, respectively, and i/y and Gj, the Poisson's ratio and the shear modulus in the {e u e,) plane, respectively. Note that, for the special case of isotropy, only two material parameters E and v are needed: E\ = E 2 = E 3 — E , v\2 — v-n = "is = v , Gi 2 = G» = G a = 2(j^TJ) • Since matrix C^'J of elastic moduli is associated with the principal material direc- tions, we need to transform it from the lamina coordinate axes {ai, a 2 , a 3 } to the global Cartesian coordinate axes {bi, b 2 , b 3 }. With 6 being the fiber direction angle relative to the global Cartesian system (see Figure 3.4), the relationship between the lamina coordinate system and the global Cartesian coordinate system is given by a x = cos Obi + sin 6b 2 , a 2 = - sin 9bi + cos 8b 2 , a 3 = b 3 . (3.88) Since we are developing the formulation in the convective coordinates associated with the basis {G t }, we have to express the tensor C of elastic moduli in the same convec- tive coordinates. Thus, C = C abcd a a ®a b ®a c ®a d = C m G { ® Oj ®G k ®d, (3.89) where the components C abcd are given in (3.86), and the components C ijkl are to be com- puted for use with the present solid-shell formulation. From the following component forms of the second Piola-Kirchhoff stress tensor S S = S ab a a <g> a b * S lj Gi ® G j , (3.90) 92 we obtain the relation between the components S ij and S ab as S l > = (&-a a )(&-a b )S ab , (3.91) where G l 'Gj = $> jt and a* = a\ Similarly for the Green-Lagrange strain tensor E, E = E cd a c ®a d = EyC? (g) G j , (3.92) we obtain the following relation E cd = (G k -a c ) (G l -a d )E kl . (3.93) Using (3.91) and (3.93) in the following component form of the stress-strain relation (3.85) with respect to the basis {a J gab m cabcdg^ ■ (3 94) we obtain 5« - (C • a a ) (& ■ m) (<Z* - a c ) (<?' • a d ) C abcd E kl , (3.95) which when compared to the component form of (3.85) with respect to the basis {GJ & = C^ kl E kl , (3.96) leads to G«» = (C • a a ) (CP • o 6 ) (G fc • o c ) (G< • a,) &** . (3.97) The above relation can also be obtained directly by using (3.89). If we expressed (3.97) in matrix form by using the same ordering of strain and stress components described in (3.56) and (3.62), which resulted in (3.86), the constitutive matrix C ljW J in the convective coordinates associated with the basis {GJ is given by [C^] = Tq [&**] T G , (3.98) 93 with Ta = f 2 f 3 f l f 3 f 2 f 3 L 2 L 2 (3.99) (*1) 2 (t\? t\t\ {t\f (^2) (^2) ^2^2 (^2) 2£j^2 ^^1^2 ^1^2 ^1^2 ^^1^2 ^1^2 ^2^1 ^1^2 ^2^1 (4) (^1) 4^3 (^3) ^3 ^3 2t\t\ 2t\t\ t\t\ + t\t\ t\t\ + i§^2 ^2^3 + ^3^2 2t\t\ 2t\t\ t\t\ + t\t\ 2t\t\ t\t\ + t\t\ £^3 + £3^ and^ = G j * ai . 3.3.5. The ANS Method The assumed natural strain (ANS) method was originally prepared to relieve the shear locking problem that typically arises as the thickness of the shell goes to zero (MacNeal [1978] Hughes and Tezduyar [1981], Dvorkin and Bathe [1984]), and was later given a mixed variational foundation (Simo and Hughes [1986]). Here we use the ANS method to treat shear locking caused by the transverse shear strains and curvature thickness locking caused by the transverse normal strain in the present solid-shell element. 3.3.5. 1. Transverse shear strains To avoid shear locking, we adopted the ANS method as applied to the four-noded shell element in Dvorkin and Bathe [1984]. Here, a linear interpolation of the compatible transverse shear strains E[ z and Efa in (3.12), evaluated at the four midpoints A, B, C, D of the element edges, at £ 3 = (see Figure 3.5), is applied (3.100) where the coordinates of points A, B, C, D are £ A = (0,-1,0), £ B = (1,0,0), £ c = (0, 1, 0), £ D = (-1, 0, 0), respectively. The above interpolation on the transverse shear strains eliminates the shear-locking problem, and allows for pure bending deformation without parasitic transverse shear strains. 94 3.3.5.2. Transverse normal strain In the case of curved thin shell structures or in the nonlinear analysis, to circum- vent the locking effect from parasitic transverse normal strain, we employ an assumed- strain approximation for the covariant component E% 3 of the compatible Green-Lagrangian strain tensor, as done in Betsch and Stein [1995] for a stress-resultant shell formulation. Here, a bilinear interpolation of the transverse normal strains sampled at the four corners E, F, G, H of the element midsurface (Figure 3.5) is imposed, that is - (3-101) 2=1 with Ni = J (1 + $ f 1 ) (1 + £ 2 £ 2 ), and the coordinates of the corner points E, F, G, H being C = t E = (-1,-1,0), £ 2 = Z F = (1,-1,0), £ = & = (1,1,0), £ - =(-1,1,0). Figure 3.5. Eight-node solid shell element in isoparametric coordinates: Sampling points for ANS interpolations for transverse shear strains (A, B, C, D) and for transverse normal strain (E, F, G, H). 3.4. Interpolation of the Enhanced Strains In this section, we first review the regular enhanced-strain method (Klinkel et al. [1999]) and establish the optimal number of internal parameters for the enhancing strains 95 in the present solid-shell element to pass the membrane patch test and the out-of-plane bending patch test. We then propose a new efficient way to enhance the strains, and prove the equivalence of the 2-D plane elasticity elements of Simo and Rifai [1990], Taylor et al. [1976] and our new enhancing formulation. 3.4.1. The Regular Enhanced Strains Treatment To include the constant stress in the element (e), the orthogonal condition of EAS must hold in (3.20), that is J SlEdV = 0. (3.102) We define the following component forms of the enhancing strain tensor E as E = EnG* (£) ® G j (0 = t kl G k (0) ® G l (0) , (3.103) where the enhancing strain components with respect to the covectors G { (£) at any arbitrary point £ are denoted by E ijt while those with respect to the covectors G k (0) at the element center £ = by E kl . From (3.103), E {j can be expressed in terms of % kl as follows E l3 - % [Gi (£) >G k (0)] [G l (0) 'Gj (£)] , (3.104) where the covector G k (0) can be computed from the vector G t (0) by G k (0) = G*'G,(0), with [G kl ] m [Gom]' 1 and G ok i = G k (0) -G t (0) , (3.105) The matrix form of (3.104) is where the components of the enhancing strain are arranged in the same order as in (3.56), and T is the matrix that transforms the strain components relative to the basis {G t (0)} to those components relative to the basis {Gi (£)}. 96 Using the Column-matrix form {S lj } for the stress components, as in (3.62), and using (3.106), we can rewrite (3.102) as J {S^} T {E tJ } dV = J {^} r r o {fyW-0. (3.107) For constant stresses, we have the following condition on j-Ey j J {%j}dv m £ £ £ jdedede = o , (3.108) 44 where J is the determinant of element Jacobian matrix of the mapping from the isopara- metric space □ to the initial configuration Bq ] of element (e). Let j^yj be defined by using the interpolation matrix M and the element parameter as follows {£y}.= ±M(0a«. (3.109) Substituting (3.109) into (3.106), the enhancing strain {^V,} can be written as = with£ = ir M. (3.110) Remark 3.3. In Simo and Rifai [1990] and other papers such as Klinkel et al. [1999], the matrix Q involves the calculation of the determinant J Q of the Jacobian matrix evaluated at eh element center, that is g = jT M. (3.111) From our numerical experiments, both expressions for Q (without J as in (3.110) and with J as in (3.1 1 1)) led to exactly the same results. We therefore use only (3.1 10) for computational efficiency. I If we only enhance the membrane strains [E n , E22, 2E i2 ], and the transverse normal 97 strain E 33 , the transformation matrix T in (3.106) should be presented as follows Tn = ■ W) 2 {a\f a\a\ (<4) 2 {alf a\a\ {alf 2a\a\ 2a\a\ a\a\ + a\a\ 2a\a\ {a\f a\al {alf (3.112) where the coefficients a{ are evaluated by a? = G f (£)•<?>' (0) , i,j = 1,2,3. (3.113) The interpolation matrix M should be constructed to satisfy (3.108) for arbitrary matrix a^. The selection of M is not unique. In the present solid-shell element, the matrix M with the minimum internal parameters of five is in the form of M = o e, 2 o o e 1 <e 2 o £ 3 (3.114) which is the same as that used in Klinkel et al. [1999]. Our numerical experiments showed that the selected M as in (3.114) cannot pass the out-of-plane bending patch test, while passing the membrane patch test. Remark 3.4. The concept of patch test was first introduced by Bazeley, Cheung, Irons and Zienkiewicz [1965] and has since demonstrated to give a sufficient condition for convergence (e.g., Irons and Loikkanen [1983], Taylor, Simo, Zienkiewicz and Chan [1986], Zienkiewicz and Taylor [1997]). A reviewer pointed out that there is no consensus about the necessity of passing the out-of-plane bending patch test for convergence, while passing the membrane patch test is necessary for convergence. On the other hand, we show in Section 5.4 that the solid shell formulation with Five HAS parameters, which does not pass the out-of-plane bending patch test, cannot provide accurate results for problems in- volving nonlinear material behavior (in addition to large deformation), whereas the present 98 formulation with seven EAS parameters, which does pass the out-of-plane bending patch test, provides accurate results. I To pass the membrane patch test and out-of-plane bending patch test, the bilinear polynomials for the transverse normal strain E 33 are necessary (i.e., the minimum number of EAS parameters for E 33 should be three, instead of just one as in (3.1 14)). Therefore, the optima] number of EAS parameters should be seven, as shown in the matrix M below M = t 1 £ 2 f 1 £ 2 £ 3 f 1 ^ 3 £ 2 £ 3 (3.115) A computationally more expensive choice for passing both patch tests is to include the trilinear polynomials for E 33 and bilinear polynomials for E n (Bischoff and Ramm [1997] and Betsch and Stein [1996]). In this case, the number of EAS parameters is nine, with the matrix M as shown below M = e 1 o e 2 o o e e ee o o o o £ 3 ?f £ 2 £ 3 fl£2|3 (3.116) The results of our numerical experiments showed that there is little advantage in using (3.1 16), since improvements compared to the use of (3.1 15) were insignificant. If we enhance all the six strain components [E n , 2E 22 , E l2 , E 33 , 2E 23 , 2E 13 ], the interpolation matrix M contains complete sets of polynomials up to the trilinear one, and thus corresponds to a set of 30 EAS parameters (Andelfinger and Ramm [1993] and Klinkel and Wagner [1997]). In this case, the matrix M is as shown below m=[m (1 ',m (21) ,m (22, ) m (3) ] , (3.117) 99 where the submatrices M (1) , M (21) , M (22) , and M (3) are M< 21 > M (22) = r^ 1 o " £ 2 e e o cf 3 e £ 3 . o o e e. " ee ' ee ee ee ee ee r ee s s o o o o o ee ee ee ee ee ee eee eee eee eee eee M< 3 > = The corresponding transformation matrix T previously discussed in (3.106) is now ex- pressed as follows T = W) 2 («f) 2 a\a\ (a?) 2 a\a\ 44 (<4) 2 a\a\ 44 2a\a\ 2a\a\ a\a\ + a\a\ 2a\a\ a\a\ + a\a\ 44 + a 2 a l a\a\ 44 2 a 2 a 3 2a\a\ a\a\ + a\al 2a\a\ 44 + a 3°2 44 + a 3 a 2 2a\a\ 2a\a\ a\a\ + a\a\ 2a\a\ 44 + a 3°i 44 + a 3 a i (3.118) where the coefficients a{ are the same as (3.1 13). Without a combination with the ANS method to remedy the shear-locking problem, the above 30-parameter EAS element (Klinkel and Wagner [1997]) cannot pass the out-of- plane bending patch test by itself. 100 3.4.2. Proposed Efficient Enhancing Strains In the traditional EAS method, as presented above, the 3 x 3 matrix [G ij] has to be inverted in each element so as to obtain the covector G l (0). For models of composite struc- tures that involve a lot of elements in the thickness direction, such inversion clearly adds to the computational cost. Here we propose a new method (for calculating the enhancing strain) that avoids the inversion of [C?oy], while still passing the required membrane patch test and the bending patch test. Moreover, this new method yields the same performance in terms of accuracy when compared to the traditional method that needs the covector G i (0) in (3.106), (3.112) and (3.113). Similar to (3.92), the enhancing strain tensor E (£) can be expressed with respect to either the convected basis {GJ or the Cartesian orthonormal basis {e a } as follows E = E {j G* (0 ® G j (0 = e« t k <g> e' , (3.1 19) where t k = e k , and the components e ki have the same structure as that of Ey as expressed in (3.109). With the use of the orthonormal basis {e fc }, we thus avoid the computation of the covectors G k (0) and the inversion of a 3 x 3 matrix, as mentioned above. To compute the tangent stiffness matrix, we need to compute the enhancing strain components By (£) at the Gauss points. Since our formulation is based on convective coordinates, we need to express the compatible strain tensor E c and the enhancing strain tensor E in the same convected basis, so to add these components together to form the total strain tensor E as expressed in (3.18). Once the components e ki in (3. 1 1 9) are known, the components Eq % can be computed from e kl and {Gj (£)} as follows %ii (0 = hi [t*Gi (0} [c l 'G 3 (0] , (3.120) where, unlike the use of the covectors ^ (0)} in (3.104), we do not need to invert any matrices, since the basis {e'} is orthonormal and thus e* == e\ 101 The next question is how to select the orthonormal basis |c fc } for use in (3.1 19) and for each element. For the case of fiat plates, the convected basis {GJ can be chosen to be colinear with the global Cartesian basis {ej. In this case, we simply choose the basis {ej to be the same for all elements such that ei = Ci, i= 1,2,3. (3.121) For the case of curved shell, in each element (e), we select the orthonormal basis {cj such that e 3 is colinear with the convected basis G 3 evaluated at the first Gauss point of the element (e), that is 3 - iwm ■ <3 - i22) and d and e 2 are obtained by rotating the basis vectors e x and e 2 through same rotation operation that rotates e 3 to coincide with e 3 as defined above. The computation of the rotation matrix for the above operation is given in Remark 3.5. It is noted that this rotation matrix yields directly the components of the basis vectors ei and e 2 . Remark 3.5. Given any two vectors e, t with e + t there exists a unique rotation tensor A such that e= Ae » (3.123) where A = (e«e) J 2 + e x e + _L_ ( e x e) ® (e x e) , (3.124) where the symbol ' over a vector designates the skew symmetric tensor associated with the vector (i.e. having the vector as its axial vector, for more details, see e.g., Vu-Quoc, Deng and Tan [2000]). Let e be the global Cartesian basis vector e 3 , and e be the basis vector c 3 = c 3 e t =: fet as defined in (3.122). The rotation tensor A can then be expressed in terms of the components f = e 3 as follows A = ej (g) e, such that A »ej = cj , (3.125) 102 Let a = U for easy recognition of the symbols. We have (3.126) With (3.126), (3.125) becomes A — ti ® ej = t\ei ® ej , (3.127) with A 5 With the superscript i in AJ designating the row index, and the subscript j designating the column index. We have the expression for t l z ei = fei as follows A! in terms of the components of e 3 = e^e* = 2 \2 t 3 + 1 +1 t 1 t 1+ t 3 t 1 t 2 1 + t 3 -t l 1 + t 3 t 3 + 1 + t 3 t 2 t 3 (3.128) Thus the first column in [A}] contains the components t\ of t x = c x = ^e,. The second column of A', contains the components of t 2 = e 2 = t^. 3.4.3. Equivalence Between EAS Element and Incompatible Mode Element In this section, we will show the equivalence of an EAS four-node element in plane elasticity as presented in Simo and Rifai [1990] and the incompatible-node four-node ele- ment of Taylor et al. [1976] by using the tensor form. In addition, we will also derive a new element formulation and prove that it too leads to a condensed stiffness matrix identical to that of the incompatible-mode element of Taylor et al. [1976]. The tensor form of the enhancing strain does indeed allow one to see the connection between various formulations in an elegant and simple manner. This approach has not been exploited in the literature. Note that even though Simo and Rifai [1990] stated that their element is in fact identical 103 to the incompatible-mode element of Taylor et al. [1976], they provided no proof, which is not immediately obvious, even though numerical experiments did confirm that the two elements are identical. Moreover, we have not seen any such proof of equivalence in the literature, to the best of our knowledge. In this section, we are mainly concerned with the small strain case, and thus the small strain notation e is used throughout the section. 3.4.3.1. Tensor form of enhancing strains The second-order enhancing strain tensor e can be expressed as where IE is a fourth-order interpolation tensor containing the polynomial basis functions, and a a second-order tensor containing the EAS parameters of an element. The enhancing strain e can be expressed in either the basis {G a (0)} or the basis {e a } as follows « (0 = tap (0 G a (0) <g> G" (0) = e ab (£) e a ® e b , a, /?, a, b = 1, 2 , (3. 130) where £ a/3 are the components of e with respect to the convected basis {G a (0)} evaluated at the element center, and e ab the components of e with respect to the global Cartesian basis {e a = e a }. Also, note that indices a and a in (3.130) take values in {1, 2}, since we are dealing with 2-D elements here. Using (3.130), we obtain the following relationship between components <£ a0 and components e a b of e (3.129) £«0 = £ab [e a 'G Q (0)] e b 'G (0) (3.131) or in matrix form (3.132) 104 with the strain matrix {<E Q/3 } = [g n , g^, 2e 12 | T and the matrix F„ defined below F T = (a\) 2 (a 2 ) 2 a}a 2 a 2 a 2 (« 2 ) 2 (^) 2 _ 2a\a\ 2a\a\ a\a\ + a\a\ where the coefficients a£ are obtained by a£ = G Q (0)-e^, a,/9-l,2 (3.133) (3.134) Similarly, the EAS parameter tensor a can also be expressed with respect to different bases. Here, we choose to express a with respect to the basis G m (0) ® e n and to the basis e 9 as follows a^a«G^(0)®e" = a<y®e«, (3.135) where the superscript (s) in the components aj>j, of a with respect to the basis G m (0) ® e" represents the EAS parameters in the Simo and Rifai [1990] formulation, and the super- script (t) in the components af g of a with respect to the basis ® e» represents the EAS parameters in the Taylor et al. [1976] formulation. From (3.135), the transformation between the components qW and af n is then or in matrix form «a=4«( c/,Gr m(0)) . (3.136) (3.137) where the indices (mn) and (fn) are arranged in the order {11, 22, 12, 21}, and the matrix M~ 1 take the expression M" 1 2 1 a{ a 2 a 2 , al a 2 a} a\ a 2 (3.138) 105 where the coefficients were given in (3.134). For the relation between the enhancing strain components £y and the EAS parame- ters qW, it follows from (3.129), (3.130)i and (3.135)! that the fourth-order interpolation tensor E can be expressed in component form as follows E = E^.-G* (0) <g> G j (0) ® G k (0) g e, , leading to the following component matrix equation (3.139) = #C {4?} (3.140) Similarly, for the relation between the enhancing strain components e ab and the EAS parameters af g , it follows from (3.129), (3. 130) 2 , and (3.135) 2 that the interpolation tensor E can be expressed in component form as follows E = E {t)C a d b e a <g> e b <g> e c ® e d , (3.141) leading to the following component matrix equation {eat} = {««} . (3.142) Substituting (3.137) into (3.140), then using the result in the left hand side of (3. 132), and next substituting (3.142) into the right hand side of (3.132), we obtain the relation between and E {t) as follows E {s] ki E^ M . (3.143) The interpolation matrix E {t) is chosen to be the same as in Taylor et al. [1976], and the derivatives evaluated at the element center, that is -tfy* (o) o ^y,v (o) o o £ l x#{0) o -£ 2 x $ i(o) e i x 4a (o) -e y , e (o) -ex <e (o) w (3.144) 106 where the rows correspond to the strain components {e n , e 2 2, ^12}, and the columns cor- respond to the EAS parameters a£j arranged following the order erf = {11, 22, 21, 22}. By substituting (3.144) into (3.143), and by rearranging the columns in the resulting component matrix J(W|J , we obtain 10 E {s) f 1 e e (3.145) where the rows correspond to the strain components j(£ n , <E 2 2, 2<Ei 2 }, and the columns cor- respond to the EAS parameters aft arranged following the order kl = {11, 22, 12, 21}. It can be seen that (3. 145) is the same strain-enhancing interpolation matrix for 2-D elasticity elements as suggested in Simo and Rifai [1990]. Using the tensor formalism, we can derive an EAS formulation that is different from, but equivalent to, the EAS formulation in Simo and Rifai [1990] and the incompatible element of Taylor et al. [1976]. To this end, let's express the interpolation tensor E with respect to the basis vectors {d (0)} and {G i (0)} at the element center as follows E = (0) <g> G j (0) ® G k (0) ® G, (0) . (3. 146) In parallel to the above, let's express the EAS parameter tensor a in the same basis: a = aWGi'(0)®G*(0) . (3.147) Similar to (3.135), we find the relation between qJJ and in matrix form to be {aW}=T- 1 {4?}, (3.148) where the indices (pq) and (kl) follow the order {11, 22, 12, 21}, and the matrix T -1 takes the form r „i a[ a{ a 2 2 a\ a\ a\ a? a} (3.149) Symbolic computation was used to carry out the computation in (3.143) to obtain the result shown in (3.145). 107 when the coefficient are computed as in (3.134). Since the enhancing strain j<£ij} with respect to the convected basis {Gi (0)} can be expressed as {«»} = K>r;] «>} E {s) ki K?} , (3.150) it follows from (3. 148) and (3. 150) that the relation between the interpolation matrix and the interpolation matrix E^ Pq 13 is given by E {s) (3.151) E kl that There are thus infinitely many ways to define the interpolation matrix are equivalent to each other from the tensor viewpoint. The enhancing-strain component matrix {e ab } relative to the global Cartesian basis {e a } can then be expressed in several ways {s ab } = 0«{a&}, with 0® = {tity = F» T M- 1 , (3.152) {s ab } = W {<*«}, witha (s) = F - T {jB^} = gWM, (3.153) {ea*} = a w {«a}. witha W = F - r {lE(^} = a (8) T ) (3.154) where the matrices £ (t) and (? (s) are the interpolation matrices relating to the EAS parame- ters a mn and Q £i- respectively, to the Cartesian strain components e^, and £ w is another choice of interpolation matrix. 3.4.3.2. Equivalence of condensed stiffness matrice s The matrix form of the strain tensors is used in finite-element formulation. In a typical element (e), the compatible strain matrix {e c ab } and the enhancing strain matrix {e ab } are interpolated respectively as follows {e c J = Bd< e > , {eab} = S« (e) , (3.155) (3.156) 108 where B is the strain-displacement matrix, and Q one of the interpolation matrices given in (3.152)— (3. 154), depending on the formulation used. The enhanced strain matrix {e a6 } is obtained by adding the enhancing strain {e ab } to the compatible strain {el b } {*<*} = + = B | J = [B 9] I ^ } . (3.157) The element stiffness matrix fc (e \ which is similar to (3.78), can be obtained by = J B T cBdv = J | » )c[sg]dv B M -(«) ^ ' / B T CBdV j B T cgdv ,(e) / g r CBdv / g T cgdv ,(e) ,(e) ju(e) ju(e) fc(e) fc (e) (3.158) where C is matrix of linear elastic moduli. The condensed element stiffness matrix has the same form as in the formulation by Taylor et al. [1976], Simo and Rifai [1990], and in the formulation presented in (3.146)- (3.149) fc(e) _ u{e) _ .(e) ^(e)]" 1 jr. (e) "au (3.159) For the formulation of Simo and Rifai [1990], the second term in (3.159) leads to fcto [l,(e)l"'L (<0 au J B T cg {s) dv j g^ T cg {s) dv 1* *(■=) (s) T CBdV (3.160) Substituting in (3.160) the following relation as given in (3.153) 2 g(s) (0 = git) (0 M ( (3.161) 109 where M is the inverse of M 1 in (3. 1 38), and is a constant nonsingular matrix. Then substituting (3.161) into (3.160), we obtain k, » _ J B cT cg {t) dv ft MM j Q® T CQ {t) dV (e) M~ T M T j Q [t)T CBdV = J B T cg {t) dv 1 -1 j g® T cg®dv J g {t)T CBdv , (3.162) which is exactly the same as the second term of the condensed element stiffness matrix fc (e) in (3.159) obtained from the formulation of Taylor et al. [1976]. Since the compatible part k^l of fc (e) are the same, the condensed element stiffness matrix k (e) in Simo and Rifai [1990] is exactly the same as in Taylor et al. [1976]. From (3.154), we have the following relation between and g^(i) = g^(i)T'\ (3.163) where T is the inverse of T _1 in (3. 149), and is a constant non-singular matrix. Following the same procedure as in (3.159) and (3.162), one can easily establish that the new enhanc- ing strain as given in (3. 146)-(3. 149) yields a condensed stiffness matrix identical to that obtained by Taylor et al. [1976] and Simo and Rifai [1990]. It should be noted that a formulation similar to (3. 146)-(3. 149) but evaluated at an arbitrary point £ inside the biunit cube will also produce an element that passes the patch test. In the previous section, we have indeed selected £,=£ x (i.e. the first Gauss point). 3.5. Numerical Examples The finite element formulation of the present low-order solid-shell element for static analyses of multilayer composite shell structures, presented in the previous sections, has been implemented in both Matlab and the Finite Element Analysis Program (FEAP), devel- oped by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a Compaq Alpha work- station with UNIX OSF1 V5.0 910 operating system. The tangent stiffness matrix and the 110 residual force vector are evaluated using full 2 x 2 x 2 Gauss integration in each element. A tolerance of 10" 18 on the energy norm is employed in the Newton iteration scheme for the convergence. Below we present numerical examples involving geometrically nonlinear static analysis, with isotropic and orthotropic elastic materials. 3.5. 1 . Patch Tests and Optimal Number of Parameters The patch tests for the membrane behavior and the transverse out-of-plane bending behavior of plate and shell elements were suggested by MacNeal and Harder [1985]. In these tests, a patch of five plate/shell elements with four external nodes and four internal nodes at the (X,Y) coordinates in Figure 3.6. (see also MacNeal and Harder [1985]): Since we are interested in submitting the solid-shell element formulation described in the previous sections to the above patch test, the number of nodes is actually doubled, with one series of nodes at the top surface of the plate (Z = h/2), and one series at the bottom surface (Z = -h/2). The aspect ratio is defined by the triplet A := L : W : h, where (L, W, h) designates the length, width, and thickness of the plate, respectively. We will subject the following elements to the above mentioned patch tests: The proposed solid- shell element with seven EAS parameters, the solid element with thirty EAS parameters in Klinkel and Wagner [1997], the solid-shell elements with five EAS parameters in Miehe [1998/?] and Klinkel et al. [1999]. It is noted that we will discuss the EAS formulation based on the displacement gradient in details in Section 4.3, which is more efficient than Miehe [1998fc]. Figure 3.6. The five-element patch of a plate: Geometric dimension (L : W : h = 0.24 : 0.12 : 0.001); E = 10 6 , v = 0.25; 1, 2, 3, 4 are node numbers on top surface. O z Coord. (X,Y): 1 : (0.04,0.02) 2 : (0.18,0.03) 3 : (0.16,0.08) 4 : (0.08,0.08) Ill 3.5. 1 . 1 . Membrane patch test The following displacements u, v , w along the X, Y, Z, axes, respectively, are pre- scribed at the top and bottom exterior nodes of the plate: u = 1(T 3 (X + Y/2) , v = 10" 3 (Y + X/2) , w = . (3.164) The theoretical solution is a constant in-plane membrane stress field in all five ele- ments of the patch as shown below ox - o Y = 1333 , t xy = 400 . (3.165) Our numerical results show that the proposed solid-shell element, together with the element formulations by Klinkel and Wagner [1997], Miehe [1998ft], and Klinkel et al. [1999] all pass the membrane patch test, that is, the computed displacements at the interior nodes agree exactly with (3.164), and the computed stresses in all elements agree exactly with (3.165). 3.5. 1 .2. Out-of-plane bending patch test To construct a constant stress state, the displacements ( u , v , w Q ) and rotations (Ox , Y )at the midsurface of the plate (i.e., Z = 0) should be u = v = , w = 10- 3 [X 2 + XY + Y 2 ) /2 , e x = 10" 3 (Y + X/2) , 6 Y = -10- 3 (X + Y/2) . (3.166) With the deformation as prescribed by (3.166), the displacements of the external nodes at the top surface and at the bottom surface of the plate can then be prescribed by [u,v,w] T = [«©, vq, u>o] T ±^[#y,-0x,O] T , (3.167) which is the prescribed boundary conditions for the out-of-plane bending patch test. The exact nodal displacements at the interior nodes are calculated by using (3.167) (see, e.g., Table 3.2 for the finite-element displacement of the interior nodes at the top 112 Table 3.2. Displacements at internal nodes in bending patch test. Present element with 7 EAS parameters, and with ANS. node number u V w 1 2.500 x 10~ 8 2.000 x 10~ 8 1.400 x 10" 6 2 9.750 x 10" 8 6.000 x 10" 8 1.935 x 10~ 5 3 1.000 x 10~ 7 8.000 x 10~ 8 2.240 x 10~ 5 4 6.000 x 10~ 8 6.000 x 10- 8 9.600 x 10~ 6 surface). The theoretical solutions of the stresses at the top and bottom surfaces of the plate are a x = a Y = ±0.667 , r XY = ±0.200 . (3.168) Table 3.3. Displacements at internal nodes in bending patch test. Solid element with 30 EAS parameters, and without ANS. node number u V w 1 1.15827 x 10" 7 2.34981 x 10" 8 5.44389 x 10~ 6 2 1.40031 x 10" 8 8.58585 x 10" 8 2.51957 x lO" 5 3 1.81490 x 10~ 8 6.05335 x 10~ 8 2.89471 x 10~ 5 4 1.37490 x 10- 7 4.70933 x 10~ 8 1.65964 x 10" 5 Without the use of the ANS method on the transverse shear strains, the solid element even with the full 30 EAS parameters (Klinkel and Wagner [1997]) cannot pass the out- of-plane bending patch test, that is, it cannot obtain the same nodal displacements shown in Table 3.2, and the stress state shown in (3.168). The calculated displacements at the interior nodes on the top surface for the solid element with 30 EAS parameters are listed in Table 3.3. Some researchers used the ANS method in combination with the EAS method with one parameter for transverse normal strain £33 (e.g., Miehe [19986], Klinkel et al. [1999]). Here, we show, however, that this formulation cannot pass the out-of-plane bending patch test (Table 3.4), while the computed results are much better than 30-parameter EAS solid element (Klinkel and Wagner [1997]). These results indicate that the ANS method plays an effective role for remedying shearing locking in thin-shell problem. 113 In the present work, we propose an optimal formulation for EAS treatment, that is, the minimal number of parameters that is required for the element to pass both the mem- brane patch test (easy) and the out-of-plane bending patch test (more difficult), in which we use both the ANS method and a seven-parameter EAS method (four parameters for the membrane strains (E n , E 22 , 2E 12 ), and three parameters for the for transverse normal strain E 33 ). It is noted that the three EAS parameters for the transverse normal strain £33 correspond to a polynomial with one linear term £ 3 and two bilinear terms and £ 3 <!; 2 . Numerical results show that this solid-shell element formulation passes the out-of-plane bending patch test, namely, obtains exactly the same displacements at the interior nodes as shown in Table 3.2, and the stresses at the top surface and at the bottom surface as shown in (3.168). Bischoff and Ramm [1997] used a nine-parameter EAS method to treat the incom- pressibility problem (five parameters for the membrane strains, and the full four parameters for the transverse normal strain). It turns out that this choice also passes the membrane and the out-of-plane bending patch tests, but with higher computational effort, compared to the seven EAS parameters that we are proposing. In this sense, the combination of the ANS method and the proposed seven-parameter EAS method (four for membrane strains, and three for transverse normal strain) is computationally optimal. Table 3.4. Displacements at internal nodes in bending patch test. Solid-shell element with 5 EAS parameters, and with ANS. node number u V w 1 2.49406 x 1(T 8 2.00207 x 10~ 8 1.39803 x 10" 6 2 9.74938 x 1CT 8 5.94341 x 10- 8 1.93334 x 10~ 5 3 1.00572 x 10- 7 8.01161 x 10" 8 2.23496 x 10~ 5 4 5.93407 x 10- 8 5.97675 x 10- 8 9.55657 x 10" 6 3.5.2. Cantilever Plate We use a single-layer cantilever plate to establish the correctness of the present for- mulation, by comparing the computed results to those published in the literature (e.g., those 114 from geometrically-exact shell theory (Vu-Quoc, Deng and Tan [2000])). 3.5.2. 1. Cantilever beam: in-plane bending This problem has been previously investigated by Simo et al. [1990] to show the superior performance of their mixed finite element shell formulation for the membrane behavior based on the Hellinger-Reissner functional. A cantilever beam subjected to an end load is discretized with ten elements. The first mesh contains elements with uniform and regular geometry (Figure 3.7), whereas the second contains highly distorted elements (Figure 3.8). The beam has length L = 1.0, width W - 0.1, and the thickness h = 0.1. The material properties are £ = 1.0xl0 7 , ^ = 0.3, (3.169) where E and v are the Young's modulus and the Poisson's ratio, respectively. The load deflection curves for both meshes are shown in Figure 3.9. The present solid-shell element based on the Fraeijs de Veubeke-Hu-Washizu functional shows the same accurate results and insensitivity to mesh distortion as for the element reported in Simo et al. [1990]. A Figure 3.7. In-plane bending: Deformed and undeformed regular mesh with 10 solid-shell elements. 115 Figure 3.8. In-plane bending: Deformed and undeformed distorted mesh with 10 solid- shell elements. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Displacement Figure 3.9. In-plane bending: Load deflection curve for both regular mesh and distorted mesh with 10 load steps. 3.5.2.2. Cantilever plate: out-of-plane bending A cantilever plate of length L = 10 and width W = 1 is subjected to the transverse shear loading F at the free end. We consider three different values of the plate thickness 116 h(l, 0.1, 0.01), which correspond to three different aspect ratios L/h(10, 100, 1000), re- spectively. Ten elements are used to model this problem (Figure 3.10 and Figure 3.12). We use various different aspect ratios to test the performance of the present solid-shell element. To have the same level of deflection magnitude regardless of the thickness h, the applied loading F is set to be proportional to the thickness raised to power three (i.e., h 3 ) in the numerical examples in this section. The material properties are prescribed to be £=1.0xl0 7 , i/ = 0.4, (3.170) where E and v are the Young's modulus and the Poisson's ratio, respectively. Figure 3.10. Out-of-plane bending: Geometry and mesh of cantilever plate. First, we point out the importance of using a combination of EAS and ANS methods in the present element, by solving the linear problem with the transverse loading F = 10 4 h 3 at the free end. A comparison of the tip deflection of the plate for different options in the present solid-shell element and beam theory is shown in Figure 3.11, where all results are normalized to the current solid-shell element. There are differences in the results obtained from elements with EAS and from elements without EAS (purely displacement formulation): the results show that the EAS method clearly improves the bending behavior, particularly for low aspect-ratio structures; its influence diminishes dramatically, however, as the aspect ratio increases. The ANS method, on the other hand, remains important throughout the large range of aspect ratio, from low to high. For aspect ratio larger than 100, the ANS method plays a more important role than the EAS method. But the ANS method alone cannot provide accurate results, when compared to the exact solutions for beams, 117 0.8 g 0.6 0.4 0.2 -£.----8 1 fr fr 1 ! -O disp -*- EASonly -+- ANS only -9- EAS&ANS -0- Timoshenko beam — E-B beam \ \ \\ w W io ! 10 L/h 10' 10' Figure 3.11. Out-of-plane bending: Relative importance of EAS and ANS method in re- lieving shear locking. as shown in Figure 3.1 1. A combination of both EAS and ANS methods is important to achieve accurate results, regardless of the aspect ratio. In Figure 3. 1 1 , the small discrepancy between the exact solution for the Timoshenko beam and the present sold-shell formulation (with EAS and ANS) is due to the effect of the Poisson's ratio. If the Poisson's ratio is set to zero, the present solid-shell formulation produces results that agree with the Timoshenko beam theory. For flat plates undergoing small deformation, numerical results show that it is suffi- cient to consider the ANS treatment for only the transverse shear strain E n and E 23 ; the additional ANS treatment for the transverse normal strain £33 does not change the numer- ical results. For the geometrically nonlinear problems, we applied the tip loading in five load steps to reach the total force F = 5 x 10 4 /i 3 . The free-tip transverse displacement along the force direction at the corner of the midsurface of the plate agrees well with the tip displacement obtained from the geometrically-exact shell element (e.g., Vu-Quoc and Tan 118 [2002a]) in which the selectively reduced integration was employed; see Table 3.5. Even in the extremely thin plate case (aspect ratio = 6667 or h = 1.5 x 10~ 3 ), the present solid-shell element yields excellent results, without any sign of shear locking. Table 3.5. Out-of-plane bending: Tip deflection of cantilever plate for wide range of aspect ratios. Comparison between proposed solid-shell and geometrically-exact shell. Aspect ratio Ljh Present element Geometrically-exact shell element 10 7.5083 7.4897 100 7.4146 7.4144 1000 7.4137 7.4137 5000 7.4140 6667 7.4139 To verify the coarse-mesh accuracy of the proposed solid-shell formulation, we use the numerical solution obtained from using 640 geometrically-exact shell elements (80 elements along the length of the plate, and 8 elements along its width) as the reference solution. The results tabulated in Table 3.6 show that even the coarsest mesh of two solid- shell elements can capture the geometrically nonlinear response with good accuracy. In general, the results obtained from the present solid-shell formulation agree well with those from the geometrically-exact shell formulation. Table 3.6. Out-of-plane bending: Convergence of computed solution, for h = 0.01. Mesh (elem. aspect ratio) geometrically-exact shell present (relative error (%) ) 2 x 1(500) 7.61149 7.61355 (5.6) 10 x 1(100) 7.41366 7.41366 (2.8) 20 x 2(50) 7.15862 7.16073 (0.7) 40 x 4(25) 7.19335 7.19744 (0.2) 80 x 8(12.5) 7.20918 7.21180 (0.0) To check the conditioning of the tangent stiffness matrix of the proposed solid-shell element, we present in Table 3.7 the number of Newton iterations for each load step and the total number iterations for five load steps for three different shell formulation: The present solid-shell formulation, the four-node solid-shell of Bischoff and Ramm [1997] 119 F Figure 3.12. Out-of-plane bending: Undeformed and deformed mesh with 10 solid-shell elements. (see its implementation in the Appendix A.l), and the geometrically-exact shell element (Vu-Quoc, Deng and Tan [2000]). For thick and moderately thin plates (10 < aspect ratio < 200), the three shell element formulations have a similar performance. For thin plates (aspect ratio > 500), the geometrically-exact shell element formulation provides the best performance. The result indicates that the kinematic description of the geometrically-exact formulation, in which the displacement of a reference surface and the finite rotation of a transverse fiber are unknown kinematic quantities, leads to a better-conditioned tangent stiffness matrix than that of the solid-shell formulation, but with more complex operation (Vu-Quoc, Deng and Tan [2000], Vu-Quoc et al. [2001]). 3.5.3. In-plane Bending Problem with Nearly Incompressibility As noted before, the displacement-based formulation exhibits severe locking when using the full 3-D constitutive models in the incompressible limit. Here we demonstrate that the proposed solid-shell element is able to alleviate the parasitic phenomenon. The beam is clamped at one end and subjected to an in-plane bending moment at the 120 Table 3.7. Out-of-plane bending: Number of iterations in each load step and total number of iterations in five load step. aspect ratio present solid-shell 4-node solid-shell geometrically-exact shell 10 9,10, 8, 7, 7(41) 9, 10, 9, 8, 7(43) 8, 9, 8, 8, 7(40) 100 12,13,10, 9, 9(53) 12,13,11,10,10(56) 11,15,10, 10, 9(55) 200 13,13,12, 9, 9(56) 13,14,13,10,10(60) 12,14,11,10,11(58) 500 20,15,16,10,10(71) 20, 17, 20, 12,10(79) 13,14,12,10,10(59) 1000 25,16,15,10, 10(76) 26,21,17, 13,11(88) 15,16,15,11,10(67) other (Figure 3.13). The material properties are £ = 4.0xl0 3 , v = 0., 0.25, 0.49999, (3.171) where E and v are the Young's modulus and the Poisson's ratio respectively. The beam is modeled with 10 x 4 x 1 finite element mesh with the plane strain constraint (constraining the thickness change). The results of this linear problem are listed in Table 3.8. The results from the standard displacement formulation, and the classical B-bar element (Nagtegaal, Parks and Rice [1974], Simo and Hughes [1986]), and the exact solution from the Euler-Bernoulli beam theory for v — are given for the comparison. 10 M = 2.75 0.25 Figure 3.13. In-plane bending problem with nearly incompressibility: Geometry and mesh. Table 3.8. In-plane bending problem with nearly incompressibility: Deflection at center of free end V Displacement B-bar Present Exact 0.1834 0.1868 0.2063 0.20625 0.25 0.17434 0.1810 0.1921 0.49999 1.591E-04 0.1453 0.1456 It can be seen that the present solid-shell element performs very well for all cases, 121 including the incompressible limit, while the displacement formulation and B-bar element lags behind for small values of v. The displacement formulation locks severely for the nearly incompressible limit in plain strain case. This indicates that for problems where the thickness stretch may be constrained (contact or external surface loading) and the in- compressible constraints (e.g., plasticity) be involved, the present element will produce a reliable results. 3.5.4. Snap-through of a Shallow, Cylindrical Roof under a Point Load This example illustrates the use of the arc-length method (e.g., Simo, Wriggers, Schweizerhof and Taylor [1986] and Schweizerhof and Wriggers [1986] for the implemen- tation in FEAP) to obtain the unstable static equilibrium response of an elastic shell struc- ture that exhibits snap-through behavior. The shell in this case is a shallow, cylindrical roof, pinned along its straight edges and loaded by a point load at its midpoint. The dimensions of roof and material properties are shown in Figure 3.14. The roof is assumed to deform in a symmetric manner, so that one quadrant is discretized, as shown in Figure 3.14. A regular 4x4 in-plane mesh of solid-shell elements are used, and two elements are used in the thickness direction since the hinged boundary need to be prescribed at the middle line of the straight edge of roof. In the R/h = 200 case, the load-deflection path contains two Figure 3.14. Snap-through of a shallow, cylindrical roof under a point load. A 4 x 4 x 2 mesh is used for one quarter of the panel, with symmetric boundary conditions. Two cases are considered by varying the thickness: R/h = 200 and R/h = 400. limit points, and the displacement control can be used to solve the equations successfully (left of Figure 3.15, with 30 load steps). In the R/h = 400 case, the path, however, "kicks 122 in" after the first limit point, the arc-length control has to be used (right of Figure 3.15, with 36 load steps). The results shown in Figure 3.15 are in close agreement with those reported in Rifai [1993, p.245], and Example 4.2.6 of ABAQUS [1995]. present at central □ snell element p-esent a: sag* Q shell element Displacement (mm) Displacement (mm) Figure 3.15. Snap-through of a shallow, cylindrical roof under a point load. Load- deflection path for the R/h = 200 case, where displacement control is employed (left), and load-deflection path for the R/h = 400 case, where arc-length control is employed (right), both compared to the geometrically-exact shell element. 3.5.5. Pinched Hemispherical Shell The pinched hemispherical shell can be considered as one of the most severe (and meaningful) benchmark problem for nonlinear analysis of shells (Stanley [1985]). The undeformed configuration of the hemispherical shell has an 18° hole at the top (North Pole), and is subjected to two inward forces at 0° and at 180° longitude on the equator, and two outward forces at 90° and 270° longitude on the equator, respectively (see Figure 3. 16). The material and geometric properties are E = 6.825 x 10 7 , u = 0.3 , R=10, h = 0.04 , (3.172) where E and v are the Young's modulus and the Poisson's ratio, R the radius, and h the thickness of the shell, respectively. Because of symmetry, only one quadrant of the shell is modeled (Figure 3.16). The computed displacements for the small-deformation case along the direction of unit loads 123 are listed in Table 3.9. For four different meshes with increasing number of elements, the values of displacements are normalized with respect to the converged value of 0.094 (MacNeal and Harder [1985]). The performance of the present solid-shell element is quite remarkable, compared to the geometrically-exact shell element. Table 3.9. Pinched hemispherical shell: Normalized displacement for linear small defor- mation. Node per side present element geometrically-exact element 3 1.083 0.909 5 1.040 0.993 9 1.003 0.987 17 0.995 0.988 For the large-deformation case, we choose a smaller thickness of h = 0.01 and the radius-over-thickness ratio R/h = 1000. The same problem was considered by Parisch [1995] for investigating the behavior of several types of elements in thin-shell applications. The mesh is composed of 16 x 16 x 1 solid-shell elements. The total load is applied in fifteen equal steps. The final deformed mesh configuration is shown in Figure 3.18 without any magnification of the deformation. A plot of the pinching loads versus the deflections at the corresponding pinching points is shown in Figure 3.17, by comparing with the 4- node degenerated shell element in Parisch [1995]. From Table 3.10, it is observed that the present solid-shell element is somewhat better than the four-node degenerated shell element in Parisch [1995], when both are compared to the converged results of eight-node degenerated shell element in Parisch [1995]. Table 3.10. Pinched hemispherical shell: large-deformation displacements due to pinched force F = 5. Element type u at B u at C 4-node shell elem. (Parisch [1995]) 3.24803 5.43434 present solid-shell elem. 3.26055 5.48331 8-node shell elem. (Parisch [1995]) 3.32798 5.84238 124 Figure 3.16. Pinched hemispherical shell: One quadrant of hemisphere with 18° hole. 3 ] 1 r 1 1 point B (outward) — point C (inward) o x o Parisch [1995] x Parisch [1995] / / / / / / / / P * / / / / 1 / / / d x / / / / / / / / / /* / ✓ s or ^x- * " i i s s 1 1 1 )L^=^_ 1 1 1 1 1 — 0.1 0.2 0.3 0.4 0.5 Normalized displacement, u/R Figure 3.17. Pinched hemispherical shell: Load deflection curve of the nonlinear calcula- tion. 125 C B Figure 3.18. Pinched hemispherical shell: Deformed hemisphere at F/2 = 2.5, viewing through hole. 3.5.6. Multilayer Composite Plate While the results in the previous section are restricted to single-layer shell, we now provide numerical examples related to multilayer composite shells in both liner and non- linear deformation regimes. 3.5.6. 1 . Two-layer composite plate: linear solution Consider a two-layer laminated plate with angle ply (±9) construction (Figure 3.19, 0° along axis X), are clamped on all sides, and subjected to an uniformly distributed trans- verse downward load on the top surface. The side length of the square plate is a = 20.0, the layer thickness y)h = 0.01, and the total thickness h = 0.02. The magnitude of the uniformly distributed load is q = 1.0. The layer material properties are E n = 40 x 10 6 , E 22 = E 33 = 10 6 , fi2 = ^13 = ^23 = 0.25, (3.173) G\2 — Cn3 = £?23 = 0.5 x 10 6 , 126 Because of fiber-orientation induced stretching/bending coupling, which eliminates the symmetry condition found in single-layer homogeneous plates, the entire plate has to be modeled. A mesh of 6 x 6 x 2 element, with one element per layer in the thickness direction, is used. In Table 3.11, the transverse displacement of the plate center is compared to both the series solution given by Whitney [1969] and the computational results obtained with a high-order hybrid multilayer shell element with the same in-plane mesh in Spilker and Jakobs [1986]. The present solutions are more accurate than those in Spilker and Jakobs [1986] , when taking the series solution of Whitney [1969] as reference. Both sets of nu- merical results (Spilker and Jakobs [1986] and Whitney [1969]) show an increased relative error when compared to the series solution as the ply angle 9 decreases, with our solution being always closer to the series solution of Whitney [1969]. The magnified deformed configuration of the two-layer composite plate is shown in Figure 3.20. Table 3.1 1. Two-layer composite plate: linear transverse displacement at plate center. Angle ±9 series solution Spilker et al. present element relative error (%) ±45 57.80 58.58 58.92 1.95 ±35 55.26 56.88 56.75 2.71 ±25 47.10 51.44 50.22 6.62 ±15 33.82 40.18 38.15 12.8 To test the performance of the present solid-shell element for high aspect ratios by decreasing the plate thickness, we decrease the magnitude of the loading as the cube of the plate thickness, so that the transverse displacement at the plate center remains the same in the series solution. The results compiled in Table 3.12 show that the computed linear solutions are accurate for a large range of aspect ratios, for the ply angle 9 = ±45°. 3.5.6.2. Multilayer composite plate with ply drop-offs This example demonstrates the applicability of the present solid-shell formulation to analyze composite structure with ply drop-offs; an example of such structures would be a (composite) plate with piezoelectric patches at the top or bottom surface. 127 Figure 3.20. Two-layer composite plate: Deformed shape with solid-shell elements. Table 3.12. Two-layer composite plate: Transverse displacement at plate center for large plate aspect ratios, with ply angle ±45. Layer aspect ratio (a/ (e)h) series solution present element relative error (%) 1000 57.80 58.92 1.95 10000 57.80 58.91 1.93 20000 57.80 57.54 0.44 In this example, each layer is made of unidirectional fiber-reinforced material, with the fiber directions aligned at 45/-45/45/-45/45/-45 degrees with respect to the length di- rection (Figure 3.21, 0° along axis X). The plate, with length L = 12 and width W = 6, has a total of six layers at the thick end, which is clamped; the free thinner end is sub- jected to a transverse normal load distribution uniformly along the free edge. The location of the ply drop-offs are at X = 4 and X = 8 with the top two layer removed after each drop-off. The layer material properties are E n = 25 x 10 9 , E 22 = E 33 = 10 9 , u n - fia = ^23 = 0.2 , G n = Giz = G 23 = 0.5 x 10 9 . Three different values of the 128 thickness of any given layer (all six layers have the same thickness) are considered, that is, (i)h = 0.1, 0.01, and 0.004, for I = 1, 6, where (£) represents the layer number. The FE mesh is composed of 288 elements with 12 elements along the length, six elements along the width, and one element for each layer through the thickness direction. The applied load on the free tip is increased in five load steps up to the total force F = 6 x 10 9 (t)h 3 , which is proportional to the cube of the layer thickness. The computed free- tip transverse displacement along the force direction at the corner of the bottom surface of the plate are presented in Table 3.13. It is observed that unlike the isotropic plate in Subsection 3.5.2.2, the level of deflection magnitude is not proportional to the cubic of the thickness in this nonlinear composite plate problem. The deformed plate is shown in Figure 3.22. Figure 3.21. Multilayer composite cantilever plate with ply drop-offs: Undeformed mesh with = 0.1. Table 3.13. Multilayer composite cantilever plate with ply drop-offs: Nonlinear transverse displacement. Layer aspect ratio Lj ((e)h) Transverse disp. 120(0.1) 6.72325 1200(0.01) 6.11453 3000(0.004) 6.01374 To test coarse-mesh accuracy, we consider the plate with layer thickness mh = 0.01, and use the computed solution obtained from the FE mesh with 24 elements along the length and 12 elements along the width as the reference solution. From Table 3.14, it can be seen that the coarse mesh with six elements along the length and three elements along 129 the width already captures the geometrically nonlinear response with a great degree of accuracy. Table 3.14. Multilayer composite cantilever plate with ply drop-offs: Performance at coarse mesh, (t)h = 0.01. Mesh (elem. aspect ratio) present element relative error (%) 6 x 3(200) 6.03514 0.02 12 x 6(100) 6.11453 1.50 24 x 12(50) 6.02229 0.00 Figure 3.22. Multilayer composite cantilever plate with ply drop-offs: Deformed mesh with w h = 0.1. 3.5.7. Multilayer Composite Hyperbolical Shell with Ply Drop-offs This examples was considered to test the current solid shell element formulation in shell structures having discontinuous geometry and strong geometric nonlinearity. The shell structure consists of three layers with the same thickness — h/3 placed sym- metrically with respect to the middle surface and two ply drop-offs at Z = 9 and Z = 15, 130 respectively (Figure 3.23). For the shell without ply drop-offs, we have compared with Basar, Ding and Schultz [1993] and the results agree each other, while a layerwise shell el- ement with complex rotation update was employed in Basar et al. [1993]. Only one eighth of the shell structure is modeled with a mesh of 14 x 14 solid-shell elements for inner layer, 14 x 10 middle layer, 14 x 6 outer layer by assuming the symmetry (Figure 3.23, 0° along circumferential direction). The layer material properties are E n — 40 x 10 9 , E 22 = E 33 = 10 9 , i/ 13 = u u = i/ 23 = 0.25 , G i2 = G13 = G 23 = 0.6 x 10 9 . The anal- ysis was carried out for three different stacking sequences: [0°/90°/0°], [90°/0°/90°] and [— 45°/0°/45°]. The deformed shapes shown in Figure 3.24 exhibit a considerable influence from the stacking sequence. The shell with ply drop-offs with the [— 45°/0°/45°] stacking sequence has larger deformation, and is less resistant to the loading than the shell with ply drop-offs with the [0°/90°/0°] and [90°/0°/90°] stacking sequences. The deformed shapes in Figure 3.25 for the final load P = 140KN demonstrate clearly that large rotations and displacements are involved in this example. Figure 3.23. Pinched multilayer composite hyperbolical shell with ply drop-offs: Unde- formed mesh. 131 so 120 100 40 60 Displacement -6 -4 -2 2 4 Figure 3.24. Pinched multilayer composite hyperbolical shell with ply drop-offs: Load- displacement diagrams, v(B) is the displacement along axis Y at point B, u(A) the dis- placement along axis X at point A. Figure 3.25. Pinched multilayer composite hyperbolical shell with ply drop-offs: Deformed shape with stacking sequence [0°/90°/0°] (left) and [-45°/0°/45°] (right). " CHAPTER 4 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER COMPOSITES : DYNAMICS 4.1. Introduction The formulation and implementation of the solid shell element for the dynamic anal- ysis of flexible multilayer shell structures undergoing large deformation and large overall motion is addressed here. In the present formulation, the dynamics of the motion of multi- layer shells is referred directly to an inertial frame, thus simplifies considerably the inertia operator. The starting point is the nonlinear dynamic weak form based on the internal en- ergy and kinetic energy of shells. A linearization of this nonlinear dynamic weak form is performed for use in the solution for the kinematic quantities via Newton's method. Time discretization is introduced to obtain the time-discrete weak form. A time-stepping al- gorithm based on the energy-momentum (EM) conserving algorithm (Simo, Tarnow and Wong [1992], Kuhl and Crisfield [1999]) is employed in the time-discrete weak form. This algorithm preserves the total momentum and the total energy exactly. Compared to the classical Newmark algorithm (Newmark [1959]), the energy-momentum conserving algo- rithm gains a more robust stability behavior. On the other hand, a high-frequency algo- rithmic dissipation is still desirable to incorporate in the EM algorithm for the stability. A number of the modifications of the Newmark family of algorithm have been proposed, typ- ically in the form of linear multi-step methods, which introduce high-frequency dissipation and preserve second order accuracy (see e.g., Hilber, Hughes and Taylor [1977], Wood, Bossak and Zienkiewicz [1981], Bazzi and Anderheggen [1982]). A small modification to the present energy-momentum conserving algorithm, in a form similar to generalized- a method (Chung and Hulbert [1993]), results in high-frequency dissipation. Numerical simulations show that the effect on the transient response is minor for small amount of 132 133 numerical dissipation. Since the kinematic description is the displacement of the top and bottom surface of the shell in the present formulation, the special treatment on the rotational degrees of freedom (Simo and Vu-Quoc [1988], Vu-Quoc et al. [2001]) is not needed. Unlike the stress-resultant shell formulation, the resulting consistent mass matrix of the present ele- ment is symmetric and independent from the configuration. The present eight-node solid-shell element relies on a new optimal (minimum) seven- parameter EAS-expansion together with the Assumed Natural Strain (ANS) method that passes both the membrane patch test and the out-of-plane bending patch test. In the for- mulation, the transverse normal strain and the membrane strains are enhanced by the EAS method; and ANS modifications on both the compatible transverse shear strains and the compatible transverse normal strain are employed to eliminate the locking effects from the compatible low-order interpolations. For the EAS approach using enhancing deformation gradient, we develop an EAS expansion by superposing the enhancing converted basis to the compatible converted basis, and then present a formulation that is much simpler than that employed in Miehe [1998^]. The presentation of the present chapter is as follows: we devoted Section 4.2 to the dynamic aspect and the use of the energy-momentum algorithm for elastic materials. A variant of the EAS formulation based on the deformation gradient (instead of Green- Lagrange strains) for solid shells is the focus of Section 4.3. In Section 4.4, several implicit direct integration methods with/without numerical dissipation are then used and compared in terms of the accuracy, stability and cost in analyses of multilayer shell structures. 4.2. Dynamics of Solid Shells by an EM Conserving Algorithm To obtain a fully discrete formulation, the temporal and spatial discretization of the weak form need to be applied subsequently. The readers are referred to Section 3.3 for the finite-element discretization of the static weak form. The classical Newmark family of implicit integration scheme (Bathe [1996]) is widely used for the temporal discretization. The Newmark algorithm is unconditionally stable for linear problems, but only condition- 134 ally stable for nonlinear problems (e.g., Simo, Tarnow and Wong [1992]). Since the stabil- ity of the time integration scheme in nonlinear problems is related to the conservation of energy, Simo, Tarnow and Wong [ 1 992] proposed the energy-momentum conserving algo- rithm in an attempt to obtain numerically stable and accurate long-term response. On the other hand, the Newmark family are symplectic and momentum preserving, and often have excellent global energy behavior with smaller time step (Kane, Marsden, Ortiz and West [2000]). In this section, we will present some results of dynamic analyses of solid shells using both energy-momentum conserving algorithm (Simo, Tarnow and Wong [1992] and Kuhl and Ramm [1999]) and classical Newmark method. 4.2.1. Time Discretization on Dynamic Weak Form N The time interval of interest [0, T] = (J [t n , i n+1 ] is subdivided in N time steps. The n=l state variables such as displacement u n , velocity u n , and acceleration u n at the beginning of the time interval [t n , t n+i ] are assumed to be known. The state variables u n+1 , u n+1 , u„ +1 at the end of the time step [t n , t n+ i] are calculated by a time-discretization algorithm. A class of time-discretization algorithms can be presented altogether as a general- ization of the Newmark method as follows. Let the inertial (mass) term be evaluated, not at the time station t n+l , but at the time t n+CXin = a m t n+ i + (1 - a m )t n , which is a point inside the interval [t n , £„+i], with the internal force and external force evaluated at another time point t n+a/ ~ a f t n+l + (1 - a f )t n inside the interval [t n , t n +i]- The time-integration parameters t n+Ctrn and t n+af have their subscript "m" and "f" being mnemonic for "mass" and "force," and are positive numbers between zero and one. The classic Newmark algo- rithm corresponds to a f = a m = 1 (Hughes [1987]); the Bossak-a algorithm corresponds to Of — 1 (Wood et al. [1981]); the energy-momentum conserving algorithm corresponds to a; = a m = i (Kuhl and Ramm [1999]). The dynamic weak form at time t n+ct; is SU (u n+af ,E n+af ) = j (SE c n+a/ : S n+a/ + 6E n+a/ : S n+a/ ) dV Bo 135 - / 6u n+aj - (b* n+af -u n+am ) pdV - J 5u n+af -t* n+Qf dS = . (4.1) Bo S„ The displacement u n+a/ at the time point t n+0lJ and the acceleration u n+otm at time point t n+am are generated by a convex combination of the corresponding quantities at time t n and time t n+ i as follows u n+af = a/u n+ i + (1 - cx f )u n , (4.2) u n + Qm = a m u n+1 + (1 - a m )5I n . (4.3) The variation 5u n+a from (4.2) <5tt„ +Q/ = ctfSu . (4.4) The velocity w n+1 and the acceleration il n+1 at time t n +\ can be expressed as a function of the displacement u n+l at £ n+1 and the computed state variables at time t n by using the Newmark method, that is l = ^2 ( U «+l - «») - - \Jp ~ 1 ] ^ n , (4.6) where and 7 are the parameters in the Newmark method, and At = t n+1 — t n the time step size. The compatible-strain tensor E° and the enhancing-strain tensor E at time t n+0tf are K+a, - & («»+«/) . (4-7) £7 B+a/ = £?(a n+a/ ) , (4.8) where the EAS parameter oc n+a/ is a convex combination of a n and a„ +1 in the same manner as that for the displacement u n+0tf as given in (4.2). For linear elastic materials, the above convex combination for the compatible strain En+a, an d the enhancing strain E n+a/ naturally leads to the identical convex combination 136 in the algorithmic stress S n+Qf as follows S n +a f = tt/Sn+1 + (1 ~ <Xf)S n = CI [a J {E c (u n+1 ) + E{a n+1 )} + (l-a f ){E c (u n ) + E(a n )]\ , (4.9) which is an essential point in energy-momentum conservation algorithm, see Remark 4.2 on the conservation of energy and momentum where the need for the definition of S n + a j becomes clearer. Substituting (4.3) into (4.1), it becomes /. T \ <^n (u n+Ctf , E n+Qf J = / (5E„ +af I S n+a/ +SE n+af : S n+a/ ^ dV + Oir pAt 2 (tin+i - u n ) + I- J 5u n+af -b* n+a/ pdV - J 5u n+af -t* n+af dS \ Bo S„ = 5U sti ff + 5U mass + SU ext = , ) pdV (4.10) where 5Tl sti ff is the virtual work generated by the internal forces, SU mass the virtual work by the inertia forces, and 6U ext the virtual work by the external forces. 4.2.2. Linearization of Dynamic Weak Form Within a typical element (e), the variation of the internal energy U stl ff in (4.10) at time t n+af is 011 stiff BP (4.11) where the variation of the strain components £?y in matrix form is given by (4.12) 137 where the strain-displacement matrix B n+0l} is a function of the discrete displacement dnla r whereas the interpolation matrix Q given in Section 3.4 is time independent. Note that the variations <fa£+ a/ = a f Sa {e) . (4.13) With (4.12), the variation of the internal energy II ^ in (4.1 1) in matrix form be- comes 511 - tdL%f% + 6* { :Z f f$ AS , (4.14) where f% ff and f$ AS are respectively the internal force related to the displacement S e) and the internal force related to the EAS parameter a^. The linearization of (4.14) with respect to the nodal displacement d^+i and internal parameter aj^ at time t n+1 , with increment Ad {e) = d^ +1 - d { *\ and Aa (e) = a (e) n+1 respectively, is then + (*2Ad« + fcW Aa W) , ( 4.15) where the stiffness submatrices fcJ^J, fcj^j, and fc£> are derived from internal forces ftug an d , whose expressions were given in (4.14). Using (4.2) and differentiate B n+aj with respect to d^ } +l , we obtain 9B n+a dB n+a dd^at = dd (e > dd^ Gaf - (4J6) Also, by using the definition of S n+af as stated in (4.9) for elastic material, it follows that HS ij } n+a } d{S*} n+1 flwW = af a» = a f CB n+i ■ and (4.17) 138 HS^} n+Qf d{SV} n+1 _ a («) af a (e) Remark 4. 1 . It is noted that in (4.18), the stress {S lj } n+af should not be computed from {Eij} n+af by using the constitutive relation S = C I E, that is {^L^^w < 4 - i9 > instead, {S tj } n+ should be computed from (4.9). I By using (4.16) and (4.17) , the tangent stiffness submatrix fe|*| in (4.15) at time t n+a/ is *S = jjgf = <*/ / (G T S„ +Q/ + B T n+af CB n+1 ) dV , (4.20) ■(«) where S>„ +Q/ is the same stress matrix as explained in Section 3.3, but evaluated at time Next, by using (4.18), the tangent stiffness submatrix fcjfj in (4.15) at time t n+af takes the form fcS = ~^ = «/ / Bl +aj CgdV. (4.21) Similarly, using (4.17), we obtain the following expression for the submatrix fc£2 in (4.15) as *&! = = a / / ifCBnHdv , (4.22) ^"+1 5 /e) and using (4.18), the submatrix k£l in (4.15) at time t n+a} as df M EAS da {e) = a f J g T CQdV . (4.23) "+ 1 B (e) It can be observed in (4.20) that the two matrices B n+Cc/ and B n+l are evaluated at two different configurations c£ e ] a/ and dj+i, respectively, it follows that the submatrix 139 fc<2> is non-symmetric. Similarly, we have feg ^ (fc£) T . The consequence is that the condensed tangent stiffness matrix kf (see (4.31)) will be also non-symmetric. From (4.10), the discrete weak form of the inertia force is Ohn ( Je) ,(e)\ _ ^L' d (e) . a " " 2 % e - A^ e)r f( fi ) ( 4 - 24 ) — UUj n+afJ mass > where the element mass matrix is obtained by using the same spatial discretization as in Section 3.3 for 6u n+af , u, u, and u: m W = J N T NpdV . (4-25) The linearization on the weak form of inertia force (4.24) is ML-*** = fe™ (e) ) Ad(e) • (426) Likewise, the contribution to the weak form (4.10) from the external forces takes the form where the element external force f^ t at time t n+aj is f% = J ^ + / • (4-28) It follows from the above expressions that the linearization of the dynamic weak form (4.10) can be written as vm%- (Ad"UaW) +D*n£L.-AdW = ^^-OTSL:-«nSS • (4-29) Substituting (4.14), (4.15), (4.24), (4.26) and (4.27) into (4.29), for any admissible 5d£ a/ and fa$a , we obtain the algorithmic tangent stiffness matrix and the force vector as follows f A^(e) 1 f -f (e) _ f (e) - f (e) iiO \ J ext J stiff J ma fc (e) ju(e) ext J stiff J mass I (A an\ 140 The condensation of the incremental EAS parameters Aa (e) in (4.30) is similar to the static case, as presented in Section 3.3. On the other hand, the non-symmetry of the tangential stiffness operator as explained above in (4.20)-(4.22), together with the addition of the inertia force f^ ass given in (4.24), have to be taken into account. Consequently, the effec- tive element dynamic tangent stiffness kft and the effective element residual force are computed from (4.30) as follows: kV = fcjfj + ,o m ^ — 4a ^aa ^au > (4-31) ,(e) f ( e ) _ f ( e ) _ f (e) , ju(e) -1 (e) fV) _ f\e) _ f{e) , ju(,e; .(e)] 1 Ae) r — Jext T stiff J mass ^ K ua [ K aa\ J EAS ' V*-J*) After an assembly process as in Section 3.3, a global matrix equation for the incremental displacement Ad is obtained. Once solved, the incremental displacement Ad is used to compute the incremental EAS parameter Aa' e ' at the element level , used for the update of the EAS parameter ot^ e \ With the converged solution d^ +l and ot£\\ obtained at time i n+1 , the acceleration and velocity d^]_ x are updated in the classical manner as shown in (4.6). Recall that the above general form of time stepping algorithm encompasses the clas- sical Newmark algorithm («/ = a m = 1), the Bossak-a algorithm (a/ = 1), and the energy-momentum conserving algorithm (a/ = a m = |). To check the conservation property of the above algorithms, the total energy and the linear and angular momenta have to be calculated. The total energy is the sum of the internal energy 8 int and the kinetic energy fC £ tot = jC + £ mt , (4.33) K = -J pu-udV , (4.34) Bo g int = -Je: SdV . (4.35) Bo where the velocity it, the strain E, and the stress S are calculated at each time step t n , and 141 so are the linear momentum C and angular momentum J C = I pudV , (4.36) Bo J = J pxxxdV , (4.37) So where x is the position vector in the current configuration, and is related to the position vector X in the initial configuration and the displacement vector u, as given by x(&t) = X(£) + u(t,t) , (4.38) similar to (3.4). Remark 4.2. In the case where otf = a m = | and b* = t* = (no external loading), energy and momentum in the system is conserved. The total energy S tot , defined in (4.33), varies within the time interval [t n , t n+ i] according to Ci - C = Kn + i -Kn + C-'i - SP ■ (4-39) From (4.2)-(4.3), it follows that U n +\ = \ («n+l + «n) , (4.40) For elastic material, that is 'E c (u) + E (a)] , (4.41) s = c : e = c : the internal energy S tnt can be rewritten as follows g int = J E , SdV = \J E ; C : EdV , (4.42) leading to the following expression for the increment of the internal energy within the time interval [t n , t n+1 ]: £n+i _ z™ 1 = 2 / (E n +i ' c I E n +i — E n : C '. E n ) dV Bo = -J (E n+1 - E n ) : C : (E n+l + E n ) dV . (4.43) Bo 142 Let the algorithmic stress S n+ i be defined as follows: S n+ i := \ (S n+1 + S n ) = \C I (E n+1 + E n ) , (4.44) From Taylor expansion, we expand the strains E n+ i and E n at time t n+ ± to get E n+1 -E n = AtE n+ x + O (At 3 ) , (4.45) Using (4.44) and (4.45), (4.43) becomes Ci - £ n* = ^ / E n+ , : S nH dV + O (At 3 ) , (4.46) Bo Likewise, it follows that u n+1 -u n = Atu n+ x + O (At 3 ) , (4.47) and hence the change of the kinetic energy K in the interval \ t n , t n +i ] is Kn+i-Kn = - Jp (u n+ i«t£ n+ i -Un* Un) dV Bo If. = -Jp (u n +i + u n ) • (tt n+1 - u n ) dV Bo = At I pu nH -u n+ i_dV + O (At 3 ) , (4.48) So Substituting (4.48) and (4.46) into (4.39), it shows ctot c tot c n+l °n J K + i + E n+ i : S n+ i] AtdV + O (At 3 ) Bo = n n+ iAr. + O (At 3 ) , (4.49) If il n+ 1 = 0, it means that the stationary condition of functional IT at time t n+ i, namely SU(u nH ,E nH )=0, (4.50) therefore the total energy is conserved in the time interval [t n , £ n +i] O-C = 0(Ai 3 ) • (4-51) I 143 4.3. Enhanced-Assumed-Strain Method Based on Deformation Gradient In this section, we will present the weak form and the finite -element discretiza- tion for the proposed solid-shell element by using the enhanced-assumed-strain (EAS) method based on the deformation gradient, instead of the reparametrization of strains in Section 3.2." 4.3.1. Weak Form The non-linear version of EAS method by Simo and Armero [1992] is based on the decomposition of the deformation gradient F into the compatible part F c and the enhanc- ing part F as follows F = F c (u) + F . (4.52) In this case, the three-field Fraeijs de Veubeke-Hu-Washizu functional, depending on the displacement field u, the enhancing deformation gradient F, and the nominal stress tensor P = FS is given by / \ n(u,F,p) = J w s (e (f c ( u ) + f)) dv - J p : Fdv So I + ijwupdV + f- J u-b*pdV - J u-t' \Bo / \ Bo S„ dS (4.53) where U sU ff corresponds to the internal energy, II mass the work by the inertia force, and U ext the work by the external forces. Remark 4.3. The naming of the stress P in (4.53) is not unique, and can be confus- ing. In component form, P is written as p = P aA e a ® Ea , such that t = P- N , (4.54) 1 1 The resulting element cannot pass the out-of-plane bending patch test while pass the membrane patch test, see the numerical examples of Section 3.5 for details. 144 where e a is the basis tangent vector in the current configuration, E A the basis tangent vector in the initial configuration, t = t a e a the traction force acting on a facet in the current configuration, and N = N A E A the normal of the same facet in the initial configuration. Truesdell and Noll [1992, p. 100], Chadwick [1976, p. 124] and Marsden and Hughes [1983, p. 135] refer to P as the first Piola-Kirchhoff stress tensor, whereas Malvern [1969, p.222] refers to T = P T , namely, the transpose of P, as the first Piola-Kirchhoff stress: T = T$ a E A ®e a , such that t = N • T . (4.55) Chadwick [1976, p.99], on the other hand, refers to T = P T as the nominal stress. The reason for the difference in the naming of this stress tensor lies in the convention adopted for the indices in the component of the (Cauchy) stress tensors (Malvern [1969, p.224]), and thus the way these tensors operate on the normal to yield the traction vector. Even though Chadwick [1976] adopted the same convention for the indices as in Malvern [1969]), he defined the traction vector as: t = P- JV = Tl • N = N -T , (4.56) and called T the nominal stress, and P = T% the first Piola-Kirchhoff stress. Here we follow the convention and naming as in Malvern [1969], that is, T = P T being called the first Piola-Kirchhoff stress, and thus call P the nominal stress. The stress power, which is related to the term / P FdV in (4.53) is written as Bo follows (Malvern [1969, p.224]) J P : FdV = / To • 'FdV , (4.57) B Bo since P : 'f = P aA (F) a A = T Aa (F) a A = T "'F . (4.58) The nominal stress P and the first Piola-Kirchhoff stress T = P T is related to the second Piola-Kirchhoff stress as follows P = Tq = FS . (4.59) 145 Remark 4.4. Regarding the functional U mass related to the inertia force, we recall that the variation of the displacement it is taken to be independent of time (see (3.21)), and thus u s = u, since Su = 0. I Using (3.12) and (4.52), the variation of the strain E is given by SE = F T SF = F T (<5F C + 5F) , (4.60) we have dW. where — — is a symmetric second order tensor (see Section 3.2). oE The weak form of (4.53) is expressed as SU (u, F, P) = SU stlff + 5n mass + SU exi , (4.62) where the weak form 6U sti ff of the internal energy II S ^ is obtained by using (4.52) and (4.61) r dW r / ~\ r dW SU stiff = J-^\ 5EdV - J S(PI F)dV = J F-^ SF c dV Bo Bo B + J F^- *. SFdV - J SP : FdV - J PI 5FdV , (4.63) Bo Bo Bo whereas the weak form 8U mass for the inertia force is simply SU mass = JSwupdV, (4.64) So and the weak form <511 &rt for the externally applied forces SRext = ~ J 5wb*pdV - J Swt*dS . (4.65) Bo S a 146 Again the nominal stress P can be eliminated from the above weak form (4.63) by en- forcing the orthogonality condition between P and the enhancing deformation gradient F, namely J 6P I FdV + JP'» SFdV = . (4.66) Bo Bo Using the definition of the notation S as in (3.28), the weak form 8U stl ff becomes 6U stlff = J SI 5EdV = I [FS : 6F C + FS : SF) dV . (4.67) Bo Bo If we develop the formulation in the convected basis, the deformation gradient F is ex- pressed by F = F c + F = g i ®G i , (4.68) where the additive decomposition of the total deformation gradient F directly leads to an additive decomposition of the convected base vectors g { in the current configuration into the compatible part g\ and the enhancing part g { , namely 9l = 9! + 9i, (4-69) F c = g1®G\ F = ^®G l . (4.70) It follows that the Green-Lagrange strain tensor E, which takes the form E = \(F T F-E 2 ) = ±(g i -9 j -G i -G j )G i ®& = \ (g\'9) + 9\% + 9i'9 C j + 9i'9i ~ GrGj) G* ® G j = E c + E. (4.71) where E = \(9t'9 c j -G i -G j )G i ®& , (4.72) E = \(9\%+g i '9) + 9 i %)G i ®G i = \9; j G i ®GP . (4.73) 147 The variation of components of the Green-Lagrange strain E is given by SEij = i (Sgi'Qj + 9 l 'Sg j ) = sym (Sg^gj) . (4.74) Substituting (4.69) and (4.74) into (4.67), the weak form 5U sti ff now reads as m stlB = J S^SE l3 dV = J S lj sym (S gi '9j) dV = J SPsym (Sg c r9j + Sg^) dV . (4.75) Bo 4.3.2. Finite Element Discretization and Linearization Following the same spatial discretization procedure as in Section 3.3, the compatible convected basis vectors g c at a point inside an element (e) in the current configuration are functions of the displacement d (e) of element (e). To pass the patch test, Simo and Armero [1992] suggested the use of the enhancing deformation gradient F tensor in the following form F = J -jF Q J TJ- Q l , (4.76) where J is the Jacobian determinant at a point £ inside an element, and J the Jacobian determinant evaluated at the element center £ = . In convected coordinates, the compat- ible deformation gradient F Q , and the element Jacobian J , both evaluated at the element center, are expressed as F Q = g c Ql ®G i , J Q = G 0l ®e\ Jo 1 = e 4 <8> , (4.77) where g oi and G 0i are given by dx(0) _ _ dX (0) 9oi ■= — . G oi •= d £ . ( 4 - 7 *) and Gq are obtained from G 0i , the vectors e { = e' form the global orthonormal basis. The tensor T in (4.76) is the same as in Simo and Armero [1992], namely 3 T = £ T; ® GRAD^/V' = T)e x ® e? 7=1,2,3, (4.79) 7 = 1 148 where Tj 6 R 3 is the vector of the internal EAS parameters a' 6 ), and N 1 the Wilson's incompatible shape function for the tri-linear brick element N 1 « J '.((*0%l) ' / = 1 - 2 - 3 - ( 4 - 8 °) The components of T\ are listed in Appendix A. 1 . Substituting (4.77) and (4.79) in (4.76), and comparing it with (4.70), the enhancing convected base g t , depending on and a^, can be expressed as with (a'") - jfaf , a t fc := | (gS-G,) . (4.81) The variation and the linearization of the convected basis vector g i and its variation bg { with respect to the displacement S e>> and the EAS parameter are given by 8 9l (d« a< e >) A 9l (d (e) ,a (e) ) Using (4.82), the variation of components of the strain tensor E takes the form SE i:i = sym (to'S*) = s Y m (fai *0j + S 9ok^'9j) + sym (g c ok S^' gj ) , (4.85) which can be recast in matrix form as {8E tj } = B (S e \ a (e) ) W (e) + 5 (d (e) , a (e) ) <W e) , (4.86) where B and B are the strain-displacement matrices associated with Sd^ and 5oS e \ re- spectively. The linearization of fcEy in (4.85) is = 5tf + Sg,, with to = tosb^i + glJT* , (4.82) = A^ + Ap^ + ^A^, (4.83) = Sg^AT? + bg&r} . (4.84) A(<5£^-) = sym[to'A&j+A(to)-^] . (4.87) 149 where Sg it Ag t , and A (Sg^ are obtained in (4.82)-(4.84). Using (4.86) and (4.75), the discrete weak form SU^ iff of the stiffness operator for element (e) is then SIl% = j {5E lJ } T {S i ^}dV = 6d^ T I B T {S^}dV + 8cx^ T J E? {&>}dV = S^ T f% + 6a^f^ St (4-88) where /^L and f ^AS are tne internal forces associated with the nodal displacement S e ^ and the EAS parameter a^ e \ respectively. The linearization of in matrix form then be written as V (SU%) • (AdW, A<xV) = 0* T + fc£Aa«) + Sa^ T (k^ u Ad {e) + k^ a Aa^) . (4.89) With the constitutive relation {S lj } = C {Eij} for the elastic material, the stiffness submatrices JfeW, kQ, and k£> in (4.89) are fcS = = / (G£A, + * T ^) dV , (4.90) * 2 = iM = S {G T uA + B T CB)dV, (4.91) 0/^ to *S2 = fff = [*£f = / (gL$u + B T CB) dV , (4.92) («) * 2 = = I {G T aa S a + B T CB)dV, (4.93) where the matrices associated with the geometrical stiffness parts in (4.90-4.93) are dB(S e \oc^) dB(d (e) ,a< e >) Gu " := dd& ' G ** :ss aSw ' c - := fldw ' C " :i= &»<■> ' (4 " 94) 150 and $„ and S Q are the stress matrices, whose expressions are listed in Appendix A.l. The linearized stationarity condition for the mixed functional IT in (4.53) is obtained by using (4.89), (4.88), and (4.65), and can be expressed in the following matrix relation ,(e) _ ,(e) -j J ext J stiff I K uu fe(e) " f Ad^ ' H fc(c) fc(e) "'act . ~Jl (4.95) EAS ) recalling that the discretization of SU ext in (4.65), when restricted to element (e) can be written as 1 OT2 = -WM-/S2, ^ (4.96) and with /j^ts ' n (4-88), we follow the same procedure for the nonlinear solution as in Section 3.3. Remark 4.5. It should be noted that in the EAS formulation based on the Green- Lagrange Strain E, while the tangent stiffness fej^ in (3.71) contains both the geometric part and the material parts, the other three tangent stiffness submatrices fej^, kg£ and fcj£| contain only the material part, because while the strain-displacement matrix B (see (3.55) and (3.73)) depends on the displacement S e \ the interpolation matrix Q does not. In the EAS formulation based on the displacement gradient F, all tangent stiffness submatrices as given in (4.90)-(4.93) contain both the geometric part and the metrical part, because both strain-displacement matrices B and B (see (4.86) and (4.94)) depend on both the displacement and the EAS parameter aS e \ I 4.3.3. Assumed Natural Strain (ANS) Treatment To treat the locking problems due to the transverse shear strains and the transverse normal strain, we consider the same approach as in Section 3.3 (i.e., the use of the ANS method). To avoid destroying the enhancement of the transverse and normal strains as already introduced in the previous section, we need to replace the compatible strain Efj in the enhanced strain by the assumed strain E(- Ns , for ij = 13, 23, 33 (i.e., the transverse and normal strains only) as follows Ekj = En - E% + E* NS , for ij = 13, 23, 33. (4.97) 151 where is the modified strain, the compatible strain Efj is given in (4.73), and the as- sumed strain E* NS is given in (3.100) and (3.101). Since the variation and the increment of the components of the compatible strain E c are SE^ = S \ (rf-flj ~ GfG,)] = sym (fcf.flj) , (4.98) A£<= = sym(A^.^ c ) , (4.99) respectively, and the linearization of the variation of the compatible strain is A = A [sym (Sg^)] = sym (Stf- Agfj , (4.100) we use the same procedure in Section 3.3 to obtain the variation 8E(- NS of the assumed natural strain and its linearization A \ SE^ NS \. The expressions in (4.85) and (4.87) are then replaced by the following 54- = SEv-dE^+SEf™ , (4.101) A(SE ij ) = A(SE ij )-A(6Et J )+A(SEf j NS ) , (4.102) for the computation of the residual in (4.88) and for the tangent stiffness submatrices in (4.90)-(4.93). 4.3.4. Simplified Formulation In the present EAS formulation based on the deformation gradient F, both the strain- displacement matrices B and B are functions of the nodal displacement S e) and element EAS parameter a (e) . Such strong coupling makes this EAS formulation more complex than the EAS formation based on the Green-Lagrange Strain E, where the strain-displacement matrix B (see (3.55)) depends only on S e \ and the interpolation matrix Q depends on neither d (e) nor a (e) . Recall that the tangent vector g { can be written as 9l = g\ (d^) + ~g x (d< e \ a< e >) , with ~ 9i = g% k (rfW) (a«) . (4.103) 152 With the Green-Lagrange strain E expressed as E=(Et j + E ij )G?<8>G i , (4.104) where compatible Green-Lagrange strain component Efj and the enhancing strain com- ponent Eij are given in (4.7 1 ). Note that g*t and thus JSy are the functions of d (e) and Recall from (3.57) that the enhancing strain in the EAS formulation based on the Green-Lagrange Strain E actually depends linearly on a (e) only. To simplify the formulation, it is possible to omit the high order term g^g^ in the expression for g*j in (4.73), that is, consider the approximation +%*«;'. (4-105) which, together with (4.82), lead to the following expression for SEij SEij = -Sgij * + -6 (gt% + g^gf) = sym (Sfrgj + ^Sfc^f '9s) + sym (dfc^f •*}) , (4.106) where the two above terms in turn lead to simplified expressions for the strain-displacement matrices B and B, respectively. We then obtain the following linearization of the variation SE of the strain = \A(6g 1J )^^A(8g^ + 1 -A(5(g c r g J +g c r g l )) = sym [5g c t - (A^ + A<4^) + Sg^-Ag] +sym (SgS-g^Aj* + Sg^A^-g') +sym (Ag&j*^ + g&ff'Ag*) , (4.107) where the above three terms correspond to the simplified geometric stiffness submatri- ces G uu , G ua , and G au associated with [Sd {e \ Ad (e) ), (<5d (e) , AqW), (<W e \ Ad (e) ), 153 respectively, while the submatrix G aa associated with (£a (e) , Ac* (e) J (similar to the ex- pression in 4.93) vanishes completely as a result of the approximation in (4.105). Readers are referred to Appendix A. 1 for details of the above matrices. We follow the same procedures described in (4.86), (4.88), and (4.89) to obtain the tangent stiffness submatrices that are similar to (4.90)-(4.93) and the internal forces similar to (4.88), and then solve the assembled nonlinear problem. Our numerical tests show that it is virtually identical for both the full formulation and the simplified formulation. The finite element formulation of the present low-order solid-shell element for dy- namic analyses of multilayer composite shell structures, presented in the previous sections, has been implemented in both Matlab and the Finite Element Analysis Program (FEAP), developed by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a Compaq Alpha workstation with UNIX OSF1 V5.0 910 operating system. The tangent stiffness matrix, the dynamic residual force vector, and the consistent mass matrix are evaluated using full 2x2x2 Gauss integration in each element. A tolerance of 10 -18 on the energy norm is employed in the Newton iteration scheme for the convergence. Below we present nu- merical examples involving geometrically nonlinear dynamic analyses, with isotropic and orthotropic elastic materials. To assess the performance of the present solid shell formulation in dynamic analyses of multilayer composite shells, we implemented four different second-order implicit inte- gration schemes, and compared their performance in relation to our solid-shell formulation. For the classical Newmark (trapezoidal) algorithm, the time-integration parameters in (4.2)-(4.6) are chosen as follows For the energy-momentum conserving algorithm, the time-integration parameters in (4.2)-(4.6) are chosen as follows 4.4. Numerical Examples (4.108) 1 2 - Poo Poo + 1 a f = Poo + 1 ' a. m 154 where p^ is the user-specified spectral radius or high-frequency dissipation coefficient (Chung and Hulbert [1993] and Kuhl and Ramm [1999]). The value p^ = 1 corresponds to the case of non numerical dissipation, while a smaller spectral radius (p^ < 1) corre- sponds to a greater numerical dissipation for the generalized energy-momentum algorithm by Kuhl and Ramm [1999]. It is noted that the numerical dissipation can also be introduced by shifting the algorithmic stress in (4.9) with a small damping parameter, as done in the modified energy-momentum algorithm (Armero and Petocz [1998]). On the other hand, both the modified energy-momentum algorithm by Armero and Petocz [1998] and the gen- eralized energy-momentum algorithm by Kuhl and Ramm [1999] lose the second-order accuracy (Kuhl and Crisfield [1999]). By simply replacing the algorithmic stress (4.9) of the average between the config- urations at the beginning and at the end of the time step with the stress at the mid-point configuration, we recover the mid-point rule from the energy-momentum conserving algo- rithm without numerical dissipation. The Bossak-a algorithm (Wood et al. [1981]) can be contained as a special case of general ized-a method (Chung and Hulbert [1993]), which is second-order accurate, and has the controllable numerical dissipation in the higher-frequency modes. The time- integration parameters of Bossak-a algorithm are 2 1 2 1 a f = l, a m = , p=- (a m ) , 7 = a m - - . (4.1 10) Poo i»4 L Although the generalized-a method possesses an optimal combination of low numerical dissipation in the low-frequency range and high numerical dissipation in the high-frequency range (Chung and Hulbert [1993]), the numerical results in Kuhl and Ramm [1999] showed that larger numerical dissipation was necessary to obtain a stable integration for nonlinear elastodynamics, when compared to the Bossak-a algorithm. 4.4. 1 . Double Cantilever Elastic Beam under Point Load This example concerns the dynamic responses of an linear elastic beam with rect- angular cross-section, built-in at both ends, subject to a suddenly applied step load at its 155 midspan (see Figure 4.1). This central part of the beam undergoes displacement several times its thickness, so that the solution quickly becomes dominated by membrane effects which significantly stiffen its response. The purpose of this example is to verify the present numerical implementation of implicit dynamic analysis. Five solid-shell elements are used Z fP = m £ = 3xl0 7 Y ©- X 20.0 v = 0.3 p = 2.54 x 10~ 4 1.0 T 0.125 Figure 4. 1 . Double cantilever elastic beam under point load. Geometry and material prop- erty. to model one half of the beam, with symmetric conditions (u = 0) applied at the midspan. The fixed time step-size is At = 50 x 10 _6 sec, and the Newmark method without numer- ical damping is used. The displacement in axis Z at the midspan along time are shown in Figure 4.2, in which the results with lumped mass matrix (obtained via (5.82) or (5.83)) is close to that from the beam element with cubic interpolation reported in Example 5.2.1 of ABAQUS [1995]. For flexural problems such as beams and shells, it is noted that the use of consistent mass matrix leads to more accurate results (Cook, Malthus and Plesha [1989, p. 375]). The linear and angular momenta along with time are demonstrated in Figure 4.3. Table 4.1, which depicts the values of the Euclidean norm of both the residual and the en- ergy norm at each iteration, clearly exhibits the quadratic rate of asymptotic convergence in the Newton's solution procedure. In nonlinear analyses, it is informative to print the energy balance, which allows us to assess how much energy has been lost. To this end, the kinetic energy, the strain energy, the total energy (kinetic + strain energies), and the work of the external forces are plotted as a function of the integration time. Figure 4.4 shows the energy values from both Newmark algorithm and the EM algorithm, where the very small energy balance error (total energy versus external work) indicates the Newmark algorithm gives the satisfactory solutions. 156 Figure 4.2. Double cantilever elastic beam under point load. Dynamic response at midspan of beam. Table 4.1. Double cantilever elastic beam under point load: Convergence results (residual norm, energy norm). Iter. Time step 1 (t=5E-5 sec) step 40 (t=2E-3 sec) step 100 (t=5E-3 sec) 4.525£+02,8.236£+00 2.009£+03, 1.804E+02 2.442£+03, 2.476£+02 1 4.479£+03,3.851£-02 2.555£+04, 1.327E+00 2.419£+04, 1.179E+00 2 3.558£-01,4.407£-10 1.551.0+01, 3.053£-05 1.029.E+01, 1.515£-06 3 1.125£-07,1.731£-22 3.917£-02,2.737£-12 4.757£-04,5.898£-16 4 4.833£-09,5.385£-26 2.218£-08, 1.536E-24 Figure 4.3. Double cantilever elastic beam under point load. Three components of linear momentum (left) and of angular momentum (right), using Newmark algorithm. 157 Figure 4.4. Double cantilever elastic beam under point load. Energy conservation using the Newmark algorithm (left) and the EM algorithm (right). 4.4.2. Pinched Cylindrical Multilayer Shell Here we simulate the large deformation of a pinched cylindrical shell with ply drop- off by using the classical Newmark algorithm. The geometry properties, material parame- ters, finite-element mesh and loading conditions for one-eighth are given in Figure 4.5. The cylindrical shell is subjected to two opposite forces acting at the mid-section and on the outer surface of the shell. The initial conditions for displacements and velocities at t = are set to zero. One-eighth of the cylinder is discretized with a FE mesh 32 x (32 + 8) solid-shell elements, with appropriate boundary conditions at the plane of symmetry. The cylinder is pinched with three different rates of loading as shown in Figure 4.5. Smaller time-step size is used for larger rate of loading, that is, 0.04 sec for Rate 1, 0.02 sec for Rate 2, and 0.01 sec for Rate 3. The deformed shape for static analysis is shown on the left of Figure 4.7, while the deformed shapes for dynamic analyses with the above men- tioned rates of loading are shown in the right of Figure 4.7 and Figure 4.8. On average, it took roughly five iterations to convergence for each time step in the dynamic analyses. Figure 4.6 displays the relationship between the magnitude of the pinching forces and the displacement at the point of application of a pinching force. The dynamic analyses clearly lead to patterns of deformation that are more complex than that obtained with static anal- ysis. Both the inertia (mass) and the loading rates have important influence on the final deformed shapes, in which buckling modes along both the circumferential direction and 158 the longitudinal direction can be clearly observed. At a higher rate of loading, larger mag- nitude of the pinching force is required for the same displacement; the overall resulting deformed shape is also more severe. Figure 4.5. Pinched cylindrical shell with ply drop-off: Geometry, material, loadings, 0° along circumferential direction. 50 100 150 200 250 300 Displacement Figure 4.6. Pinched cylindrical shell with ply drop-off: Pinching force amplitude versus displacement under pinching force. 159 Figure 4.7. Pinched cylindrical shell with ply drop-off: Deformed shape for static case (left), deformed shape at t = 2.64 for Rate 1 , time-step size 0.04 sec (right). Figure 4.8. Pinched cylindrical shell with ply drop-off: Deformed shape at t = 1 .98 for Rate 2, time-step size 0.02 sec (left), deformed shape at t = 1.5 for Rate 3, time-step size 0.01 sec (right). 4.4.3. Free-Flying Single-Layer Plate The same example as in Kuhl and Ramm [1999] is used here to compare the perfor- mance of the present solid-shell formulation to that of the eight-node shell element with reduced integration. The geometry, the loading configuration and the time history loading amplitude are described in Figure 4.9. In this example, we use thirty solid-shell elements, with a time-step size of At — 50 x 10" 6 sec and a total simulation time of t =* O.lsec. The effect of gravity force is not considered. 160 The material properties are E = 206.GPa, v = 0., p = 7800.i^/m 3 , (4.1 1 1) where E, u, and p are the Young's modulus, the Poisson's ratio, and the mass density, respectively. Snap shots of the plate undergoing large overall motion and large deformation are taken every 4 x 10~ 3 sec from the simulation by using the energy-momentum (EM) con- serving algorithm, and are displayed in Figure 4.11. Furthermore, the conservation of linear and angular momenta is demonstrated in Figure 4.12. On the left of Figure 4.16, the kinetic energy, the strain energy, the total energy (kinetic + strain energies), and the work of the external forces are plotted as a function of the integration time. Unlike the slightly increasing total energy caused by reduced integration technique in Kuhl and Ramm [1999], the total energy in the present solid-shell formulation is exactly conserved. The classical Newmark method and the mid-point rule lead to a loss of stability at an early time stage of the integration, as indicated by the dramatic increase of the total energy (Figure 4.17). The mid-point rule has a stability that is somewhat worse than that of the Newmark algorithm in this example. On the other hand, the mid-point rule conserves the angular momentum, while the Newmark algorithm conserves only the linear momentum (Figure 4.13 and Fig- ure 4.14). With a numerical dissipation set at p^ = 0.975, the Bossak-a method yields a stable integration in the whole time range (the right of Figure 4. 16), and conserves both the linear momentum and the angular momentum. The variation of the number of iterations per time-step as the integration progressed indicates a trend of increasing number of iterations and a potential for lack of convergence at some future station in the EM algorithm (left of Figure 4. 1 8). On the other hand, the Bossak-a algorithm possesses a stable number of iter- ations and thus a better rate of convergence at each time step (right of Figure 4. 1 8). While the difference in the displacements obtained from the EM algorithm and from the Bossak- a algorithm is negligible (left of Figure 4.19), the difference in the total energy is about 18% at t = O.lsec (Figure 4.16). In parallel, the difference in the velocity (right hand side 161 of Figure 4.19) is small, but the difference in the acceleration (Figure 4.20) is large. The magnitude of the acceleration obtained from the EM algorithm increases rapidly, while the one obtained from the Bossak-a algorithm hovers at around the level of 10 6 . The relative higher magnitude of acceleration suggests a higher level of "noise" in the high-frequency range of the response obtained from the EM algorithm, compared to the Bossak-a algo- rithm, whose parameter a m in (4.1 10) has a direct influence on the acceleration update as expressed in (4.3). 0.04m Figure 4.9. Free-flying single-layer plate: Geometry, loading distribution, and loading amplitude. Figure 4.10. Free-flying single-layer plate: Initial undeformed configuration at t — 0. 4.4.4. Free-Flying Multilayer Plate with Ply Drop-offs This example establishes the capability and performance of the present solid-shell in modeling multilayer plates/shells with ply drop-offs. Comparison of the stability and accuracy of the four different time-integration algorithms is provided for this example. The geometry and the mesh of a three-layer plate with ply drop-offs is shown in Figure 4.21. 162 Figure 4.11. Free-flying single-layer plate: Perspective view. 4 5 4 3.6 - 1 0.5 0.01 0.02 0.OS 0.04 0.0S 0.06 0.07 0.0» 0.0» 0.1 0.01 02 03 0.04 05 06 07 06 00 1 l" ) Tm [sec) Figure 4.12. Free-flying single-layer plate: Three components of linear momentum (left) and of angular momentum (right), using EM algorithm. The length, width, and layer thickness of the plate are the same as those of the single- layer plate in Subsection 4.4.3 (i.e., L m 0.3, W = 0.06, and {e) h = 0.001). The plate is divided into three equal parts and two ply drop-offs along its length, with each part having a length of 0.1. We use the same material properties as listed in (4.1 1 1) so that the three-layer plate with ply drop-offs here has the same weight as the the single-layer plate • -0.15 | -0.2 I -0.26 0.01 0.02 0.03 0.04 0.05 0.00 0.07 0.08 0.0» 0.1 " 0.01 0.02 03 0.04 0.05 06 07 06 0« 1 Figure 4.13. Free-flying single-layer plate: Linear momentum (left) and angular momen- tum (right) using the Newmark algorithm. 0.01 0.02 0.03 0.04 0.06 0.06 Tims (sac) -0.1 | -0.15 t -0.2 I I % -0.25 — 1 1 |— 0.07 0.08 0.09 0.1 01 0.02 0.03 DM O.OS 006 0.07 0.06 0.09 0.1 Time (sec) Figure 4.14. Free-flying single-layer plate: Linear momentum (left) and angular momen- tum (right) using the mid-point rule. - Lx - - Ly 0.01 0.02 0.03 0.04 0.05 06 0.07 0.08 09 0. Tim* : sec i -0,1 • -0 15 ■§ -0.2 I -0.25 0.01 0.02 0.03 0.04 0.05 0.06 0.07 08 09 1 Tim* (s*c) Figure 4.15. Free-flying single-layer plate: Linear momentum (left) and angular turn (right) using the Bossak-a algorithm. momen- in Subsection 4.4.3. The load distribution and the time history of the loading amplitude are also the same as Figure 4.9. A total of sixty solid-shell elements are used to model the 164 !! 'f. i'i! i I! s. i! i! i ' » V is 5 i"; h ?, i\ ti i» ! * s 1 5 2 (I W mm — Kinetic Strain Kinetic+Strain External Work * i i 5 1 MAI i m ., "' •} „ A 5 i K .n n il "i H « A A- M i 11 ■ i * n * f V -if 1/ y y U i is * 1 - Kinetic Strain Kinetic* Si rain — External Work 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0. 1 Time (sac) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Tims (sac) Figure 4.16. Free-flying single-layer plate: Energy conservation using the EM algorithm (left) and the Bossak-a algorithm (right). i--vs Kinetic Strain • Kinetlc+Straln - External Work : i : ■ ! ■ i ii ; Kinetic Strain Kinetic+Strain External Work 0.01 O.02 0.03 0.04 0.05 0.06 0.07 0.0B 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 TVna (aae) Time (aac) Figure 4.17. Free-flying single-layer plate: Energy conservation using the Newmark algo- rithm (left) and the mid-point rule (right). Figure 4.18. Free-flying single-layer plate: Number of iterations to convergence at each time step using the EM algorithm (left) and Bossak-a algorithm (right). three-layer plate with ply drop-offs. The same time-step size and time range are used as in Subsection 4.4.3. Tim* (mc) Tim* (sac) Figure 4.19. Free-flying single-layer plate: Difference in displacements (left) and in veloc- ities (right) as obtained from the EM algorithm and the Bossak-a algorithm. 1 1 1 1 1 1 1 1 EM Bossak i i i i i i i i i 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec) Figure 4.20. Free-flying single-layer plate: Difference in accelerations as obtained from EM algorithm and Bossak-a algorithm. Snap shots of the deformed shapes taken at every 4 x 10 -3 sec time interval from the computations using the Bossak-a algorithm are presented in Figure 4.22. The energy distribution for this example is given in Figure 4.28, which when compared to Figure 4. 16 reveals a smaller level of total energy in the present example of the three-layer plate in 166 relative to the single-layer plate in Subsection 4.4.3. The deformed shapes of the single- layer plate and of the three-layer plates are shown in Figure 4.23, where it can be seen that the three-layer plate has a more flexible (thinner) end and a more rigid (thicker) end when compared to the single-layer plate. Further, when comparing the initial undeformed configuration in Figure 4.21 to the deformed configuration on the right of Figure 4.23, we can see that the thick end of the plate has moved from left to right as a result of the large overall rotation of the plate. Both Figure 4.21 and Figure 4.23 were plotted using the same perspective point of view. In this example, the energy-momentum conserving algorithm could not carry on the integration beyond the time t = 56 x 10" 3 sec due to lack of convergence (right of Fig- ure 4.28) . On the other hand, the Bossak-a algorithm with the numerical dissipation Poo = 0.975 provides a stable integration in the whole time range, and conserves the lin- ear and angular momenta (Figure 4.27). With the EM algorithm, the linear and angular momenta (Figure 4.24) together with the energy (left of Figure 4.28) are conserved for the present solid-shell element up to the time t = 56 x 10 _3 sec. The difference between the displacements obtained with the EM algorithm and with the Bossak-a algorithm up to t = 56 x 10~ 3 sec is negligible (left of Figure 4.31) ; the difference in the velocity is negligible up to about t — 20 x 10~ 3 sec, and then increases to a noticeable level until t = 56 x 10 _3 sec (right of Figure 4.31). The total energy with the Bossak-a algorithm exhibits a loss of about 24%. Again the difference in the acceleration is large (Figure 4.32). The increasing number of iterations to convergence in each time step when using the EM al- gorithm (Figure 4.30) eventually leads to a numerical failure at time t = 56 x 10 _3 sec. The reason behind this increase in the number of iterations and the eventual numerical failure is the increasingly active high-frequency response in the numerical solution as manifested in the rapid increase of the acceleration (Figure 4.32). Similar to the single-layer plate in Subsection 4.4.3. The Newmark algorithm and the mid-point rule lead to a loss of stability at an early stage in the solution process, as shown in Figure 4.29. On the other hand, the loss of stability of the mid-point rule occurred earlier compared to the Newmark algorithm, 167 even though the mid-point rule conserves both the linear and angular momenta, while the Newmark algorithm conserves only the linear momentum (Figure 4.25 and Figure 4.26). Among the recent energy-momentum conserving algorithms with numerical dissi- pation, the algorithm by Kuhl and Ramm [1999] is shown to be more robust than the algorithm by Armero and Petocz [1998]. The level of acceleration in the Kuhl and Ramm [1999] algorithm plateaued out after an initial stage of increase in magnitude, and stayed that way until the end of the simulation time range (left of Figure 4.36). The acceleration in the Armero and Petocz [1998] algorithm kept increasing until a lack of convergence that prematurely halted the solution process in the same manner as encountered with the orig- inal EM algorithm by Simo, Tarnow and Wong [1992] (right of Figure 4.36). Figure 4.37 displays the number of iterations to converge in the Kuhl and Ramm [1999] algorithm and in the Armero and Petocz [1998] algorithm. The difference in displacements obtained from both EM algorithms with numerical dissipation and the Bossak-a algorithm is negligible (Figure 4.34), and the difference in the velocity (Figure 4.35) is small. Nevertheless, unlike the Bossak-a algorithm, both EM algorithms with numerical dissipation lose the desired second-order accuracy (Kuhl and Crisfield [1999]). Figure 4.21. Free-flying three-layer plate with ply drop-offs: Initial undeformed configura- tion at t = 0. From the results of above numerical examples, it is clear that the Newmark (trape- zoidal) algorithm and the mid-point rule do not guarantee a robust time integration in non- 168 Figure 4.23. Free-flying single-layer plate (left) and three-layer plate (right) with ply drop- offs: Deformed shapes at t = 16 x 10 -3 sec. . linear elastic dynamics due to their rapidly increasing energy in the integration process. By algorithmically enforcing the conservation of the total energy within each time step, the energy-momentum algorithm gains a more robust stability behavior compared to the New- mark algorithm and the mid-point rule. Yet, the EM algorithm remains unstable due to a continuous growth in the acceleration; such accelerated growth would eventually terminate 169 Figure 4.24. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu- lar momentum using the EM algorithm. Figure 4.25. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu- lar momentum using the Newmark algorithm. Figure 4.26. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu- lar momentum using the mid-point rule. the integration process due to a lack of convergence within a finite time length. Quenching the acceleration growth in the EM algorithm by introducing numerical damping in the high- 170 -0.05 ] \ ■- Jx - - Jy 01 0.02 0.03 0.04 0.05 0.06 0.07 O.OB 0.09 0.1 0.01 0.02 0.03 0.04 0.05 O.OG 0.07 0.08 0.09 0.1 Time (mc) Time (sec) Figure 4.27. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu- lar momentum using the Bossak-a algorithm with = 0.975. r, ' *i Kinetic Strain - Kinetic+Sttaln - External Work . n my ^r^ 1 jV /\ k, \ ' V * S VV''\A i S-y;-, - Kinetic Strain - - Kinetic+Slrajn — EmarnaiWork i : i 0.01 0.02 0.03 0.04 0.0S 0.06 0.07 0.06 0.0ft 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 O.OB 0.1 Time (sec) Time (sec) Figure 4.28. Free-flying three-layer plate with ply drop-offs: Energy conservation and divergence using the EM algorithm (left); energy loss and continued integration using the Bossak-a algorithm with p^ = 0.975 (right). ■-■ Kinetic Strain — Kinetic +St rain External Work 0.01 0.02 0.03 0.04 0.06 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (»ec) Time (aec) Figure 4.29. Free-flying three-layer plate with ply drop-offs: Energy balance and diver- gence using the Newmark algorithm (left) and the mid-point rule (right). I 1 1 1 1 1 1 1 1 1 1 o 1 1 1 ' 1 ' 1 1 1 ' 1 0.01 0.02 0.03 0.04 0.0S 0.06 0.07 0.08 0.09 0. 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 0.09 t T)m* (uc) Tim* (sac) Figure 4.30. Free-flying three-layer plate with ply drop-offs: Number of iterations till convergence in each time step for the EM algorithm (left) and for the Bossak-a algorithm with poo = 0.975 (right). 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0B 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 0.09 0.1 Figure 4.31. Free-flying three-layer plate with ply drop-offs: Difference in displacements (left) and in velocities (right) as obtained from the EM algorithm and the Bossak-a algo- rithm with poo = 0.975. frequency range may or may not prolong the termination of the integration process, while paying the price of losing the desired second-order accuracy. All EM algorithms, with or without numerical damping, lead to non-symmetric tangent stiffness matrices. By contrast, the Bossak-a algorithm with an appropriate amount of numerical dissipation provides a stable, and second-order accurate integration process that yields practically the same dis- placements as obtained in all other algorithms (Newmark, mid-point rule, EM algorithm with or without numerical damping) before the failure. We note on the other hand that there is a smaller loss of total energy in the algorithm of Kuhl and Ramm [1999] (left of Figure 4.33) when compared to the Bossak-a algorithm (right of Figure 4.28). 172 10 Figure 4.32. Free-flying three-layer plate with ply drop-offs: Difference in accelerations obtained from the EM algorithm and Bossak-a algorithm with = 0.975. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 05 06 0.07 0.08 0.09 0.1 TV"* (»•«) Hm« (««c) Figure 4.33. Free-flying three-layer plate with ply drop-offs: Energy balance using the EM algorithm with = 0.985 as in Kuhl and Ramm [1999] (left) and with f = 0.0001 as in Armero and Petocz [1998] (right). 173 Figure 4.34. Free-flying three-layer plate with ply drop-offs: Difference in displacements using the EM algorithm with Poo = 0.985 as in Kuhl and Ramm [1999] (left) and with £ = 0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-a algorithm with poo = 0.975. Time (uc) TJma (sec) Figure 4.35. Free-flying three-layer plate with ply drop-offs: Difference in velocities using the EM algorithm with p^ = 0.985 as in Kuhl and Ramm [1999] (left) and with f = 0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-o algorithm with Poo = 0.975. 174 1 1 1 1 1 1 — 1 EM | - - Bossak | i i i i i i 0.01 0.02 0.03 0.04 0.0S 0.06 0.07 0.0B 0.09 0.1 01 0.02 0.03 0.04 0.05 0.08 0.07 0.0B 0.09 0.1 Time (ssc) Time (sec) Figure 4.36. Free-flying three-layer plate with ply drop-offs: Difference in accelerations using the EM algorithm with = 0.985 as in Kuhl and Ramm [1999] (left) and with £ = 0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-a algorithm with poo = 0.975. 0.08 0.09 0.1 Time (sac) Time (sec) Figure 4.37. Free-flying three-layer plate with ply drop-offs: Number of iterations till convergence in each time step using the EM algorithm with p^ = 0.985 as in Kuhl and Ramm [1999] (left) and with f = 0.0001 as in Armero and Petocz [1998] (right). CHAPTER 5 EFFICIENT AND ACCURATE MULTILAYER SOLID-SHELL ELEMENT: NONLINEAR MATERIALS AT FINITE STRAIN 5.1. Introduction The analysis of general shell structures have been of interest for several decades. There is a continuing challenge to develop reliable, accurate and efficient low-order shell elements, especially for analyses of shell structures with arbitrary geometries, loadings, boundary conditions and nonlinear materials. Because of the high cost of 3-D continuum elements, shell structures are mainly mod- eled by shell elements based on either the degenerated shell concept or the classical stress- resultant shell theory. Both formulations are based on the common kinematic assumptions of inextensibility in the thickness direction and the zero-transverse-normal-stress condition. Although these approximations led to good results in most cases, several difficulties and appreciable errors could arise. Since the zero-transverse-normal-stress condition must be imposed, the implementation of 3-D material models proves to be a difficult task, and the complexity of the algorithmic treatment is increased. For example, even for the simplest von-Mises elastoplastic model, the stress-resultant constitutive models is rather complex (Simo and Kennedy [1992]). Moreover, a proper description of 2-D constitutive equations at finite strain remains a question (Schieck, Pietraszkiewicz and Stumpf [1992]). On the other hand, in many applications involving (i) the localized effects due to surface load- ings, (ii) the contact interaction of different shell structures, or (iii) the delamination of multilayer shells, it is important to include the transverse normal-stress and the associated thickness change to obtain a better accuracy (Cho, Yang and Chung [2002], Fox [2000]). Furthermore, when both shell elements and solid elements are used in one FE model such 176 as folded shell structures, 12 additional transition elements (e.g., Liao, Reddy and Engelstad [1988], Cofer and Will [1992]) or multipoint constraints (e.g., MPC in ABAQUS [2001]) are needed to connect rotational dofs and displacement dofs. solid-shell elements weld zone modeled by solid element Figure 5.1. Discretization at shell junction: Combination of solid elements and solid-shell elements. The proposed solid-shell element formulation overcomes the above mentioned diffi- culties, and improves the computational accuracy for wider shell applications. The kine- matic description of the present element consists of only displacement dofs at the top and bottom surfaces of the shell. Complex finite rotation updates such as those found in stress- resultant shell elements (Vu-Quoc, Deng and Tan [2000]). The present formulation also provides a natural way to connect to regular solid elements (see Figure 5.1) without the need for transition elements or submodeling technique as in ABAQUS [2001], in which the modeling processes are laborious and error prone. For bending-dominated problem of homogeneous shells, in contrast to the use of 3-D solid elements where a large number of layers of elements must be used in the thickness direction together with a dense mesh on the shell surface (thus leading to ill-conditioned stiffness matrices), only a single solid shell elements across the shell thickness, together with a much coarser mesh of solid shell ele- ments on the shell surface, are sufficient to provide accurate results. Unlike stress-resultant shell formulations, 3-D nonlinear complex material models can be used directly in a solid- shell, without further treatments which add complexity of the model, such as accounting for 12 It is not possible for stress-resultant shell elements to describe the detailed strain and stress distributions at shell intersections (Chroscielewski, Makowski and Stumpf [1997]). 177 the plane-stress constraint condition. Moreover, the strain-driven character of the formu- lation further simplifies the implementation of nonlinear constitutive models, in contrast to the hybrid finite-element formulations derived from the Hellinger-Reissner functional (Simo et al. [1989]). Because of the use of the covariant Green-Lagrange strain tensor without ignoring the higher order terms, all kinematic quantities such as displacements and the corresponding strains can be finite, and the update procedure can be proceeded in an exact manner, without approximations. It is noted that the quadratic terms in the strains become important in the analysis of relatively thick shells, involving strong curvatures or presenting large strains together with bending deformations (Buchter et al. [1994]). Displacement-based solid elements are known to have poor performance in bending- dominated situation and/or with incompressible materials, such as in thin shell analysis with elastoplastic material. To obtain the same performance in bending as with stress- resultant shell formulations with plane-stress assumption (e.g., Vu-Quoc, Deng and Tan [2000]), the Enhanced Assumed Strain (EAS) method and the Assumed Natural Strain (ANS) method are employed in the present solid-shell formulation. As demonstrated in Chapter 3, many solid-shell formulations unfortunately do not pass the out-of-plane bending patch test (e.g., Miehe [1998&], Klinkel et al. [1999]). To avoid the Poisson-thickness locking problem (Zienkiewicz and Taylor [1991, p. 161], Bischoff and Ramm [1997]), the transverse normal strain must have at least a linear distribution across the shell thickness. In the present formulation, an optimal number of EAS parame- ters is established to enhance both the transverse normal strain (linear distribution in thick- ness) and the in-plane strains. Moreover, the present solid shell formulation passes both the membrane patch test and the out-of-plane bending patch test, and thus with minimum computational effort. ANS interpolation has been a most successful tool to tackle the shear-locking effect in the 4-node displacement-based shell elements, even for initially distorted meshes (Mac- Neal [1978], Hughes and Tezduyar [1981], Dvorkin and Bathe [1984], Parisch [1995]), as compared to the (selectively) reduced integration. To treat transverse shear locking, we ap- 178 ply an ANS interpolation on the compatible transverse shear strains. In the case of curved structure with geometric nonlinearity, to treat the curvature-thickness locking (Bischoff and Ramm [1997]), which is also called the trapezoidal locking (Sze and Yao [2000]), we apply an ANS interpolation on the compatible transverse normal strain, as proposed by Betsch and Stein [1995]. In addition to the above features, our new contributions in this chapter are specifically listed below: 1) Demonstrate that the proposed optimal seven EAS parameters (three for the transverse normal strain to treat the Poisson-thickness locking, and four for the membrane strains to treat the in-plane bending locking) are sufficient to avoid locking problems with incom- pressible materials, in addition to passing the membrane and out-of-plane bending patch tests. 2) Justify the use of the present element with various nonlinear materials in problems in- volving multilayer composite shells, including junctions with regular solid elements and contact/impact. 3) Show that dynamic analyses can be carried out using either the consistent mass matrix or lumped mass matrix (in explicit integration 13 ) without spurious modes (Belytschko, Lin and Tsay [1984] , Zeng and Combescure [1998]). Recall that the consistent mass matrix for multilayer stress resultant shells is complex and configuration dependent (Vu-Quoc et al. [2001]). Two nonlinear 3-D material models at finite strain have been implemented in our solid-shell formulation, and the simulation results reported in the present chapter. For the (compressible) Mooney-Rivlin Model, the approach of the incompressibility limit is tackled by the use of the penalty method or the augmented Lagrangian method. Other forms of hyperelastic constitutive models such as the Ogden-type model (Ogden [1984]) are amenable to be implemented in the present element formulation. On the hyperelasto- 13 It is noted that the rotational dofs in traditional shell formulations correspond to high frequency modes, which will drive the stable time-increments to very small size, thus requiring an artificial scaling of the rotational masses (Hughes [1987, p.564]). 179 plastic model, the current implementation possesses the following advantages: (1) the re- turn mapping algorithm of infinitesimal plasticity can be carried over to the present finite deformation context without any modification, and with a simplification of the compu- tational procedure: The closest-point projection algorithm is now formulated in princi- pal stretches. In particular, the algorithmic elastoplastic moduli tensor is symmetric, and the incompressibility is automatically ensured; (2) With the elastic response emanating from the hyperelastic form of the free-energy function, the elastic predictor in the return- mapping algorithm is exact, and computed without resorting to the use of incrementally objective algorithms (Simo and Hughes [1998, (p.276)]) as for hypoelastic models; The present implementation (3) employs the consistent tangent moduli tensor, instead of the continuum elastoplastic moduli tensor, thus achieving quadratic rate of convergence in the Newton iterative procedure (Simo [1988Z?]), and (4) finite-strain elastoplasticity based on the Cauchy-Green strain tensor, thus avoiding the computation of the deformation gradient F via the costly polar decomposition. Even though there is no consensus on the necessity of passing the out-of-plane bend- ing patch test for convergence, we demonstrate that elements (with insufficient EAS pa- rameters) that did not pass the out-of-plane bending patch test perform poorly in problems involving nonlinear material behavior and large deformation, as opposed to the present formulation, which provides accurate results. 14 The outline of the present chapter is as follows. We discuss the implementation of the Mooney-Rivlin material model and the hyperelastoplastic material model, in Section 5.2. The explicit integration method for solid-shell elements is present in Section 5.3. Several numerical examples that illustrate the performance of the present formulation involving large deformation, implicit and explicit dynamic analyses, together with a comparison of the computed results to those obtained from other shell formulations and from a meshless method, are presented in Section 5.4. 14 There is on the other hand a consensus that passing the membrane patch test is necessary for convergence. 180 5.2. Nonlinear Material Law A major advantage of the present solid-shell element is that all algorithms concerning the 3-D nonlinear material models can be implemented without any modification. For non- linear materials, the second Piola-Kirchhoff stress tensor S in (3.16) and the fourth-order consistent tangent moduli tensor C in (3.70) are crucial for the numerical solution of initial- boundary-value problems with the quadratically convergent Newton-Raphson scheme. 5.2.1. The Mooney-Rivlin Material Models There are two possible ways to define the stored energy function W s for the modified Mooney-Rivlin material, in which both satisfy the zero stress condition S = at the initial configuration (i.e., E = 0). According to Fried and Johnson [1988], the stored energy function W a of a modified Mooney-Rivlin material is expressed as follows W s = d (/, - 3/ 3 1/a ) + C 2 (l 2 - 3/ 3 2/3 ) + ^ (ln/ 3 ) 2 , (5.1) where the invariants I\, I 2 , and 7 3 are expressed in terms of the right Cauchy-Green tensor C C = F T F = 2E + 1, (5.2) as follows h := trace (C) , I 2 := \ [l x 2 - trace (C 2 )] , J 3 := det (C) , (5.3) where C\, C 2 are material constants, and A the penalty parameter for incompressibility. From (5.3), the following derivatives are obtained where 1 represents the second-order identity tensor, and I the fourth-order identity tensor, both expressed in the convected basis {Gi} as follows 1 = G ij Gi ft Gj = G Ij G i ® G ] , (5.5) 1=1 (p ik G jl + G ll G ]k ) G t 9 Gj ® G k ® G l , (5.6) 181 The second Piola-Kirchhoff stress tensor S is the derivative of the stored energy function W s with respect to the strain tensor E. Using (5.1) and (5.4), it follows that (Ci + C 2 h) 1-C 2 C- (d/ 3 1/3 + 2C 2 / 3 2/3 - Aln/ 3 ) C- 1 ] , (5.7) where from (3.1 1) and (5.2) we obtain the following expressions for C and its inverse C 1 as C = F T F = g^G* ® G j , C- 1 = g ij Gi <g> G j , (5.8) respectively, with g {j and g lj computed as shown in (3.9) and an equivalent equation. The fourth-order material moduli tensor C is the second derivative of the stored en- ergy function W s with respect to the strain tensor E, that is d 2 W< dS C = OEdE dE dC~ l 4C 2 1 01- 4C 2 I + 4 (Aln/ 3 - / 3 1/3 d - 2/ 3 2/3 C 2 ) dC + 4 (A - i/3 1/3 C, - ll 2 3 /3 C 2 ) C 1 ® C" 1 , (5.9) dC~ x where the term can be shown to take the form as follows 15 dC~ l 1 -QQ- = -g (^ ifc ^' + <?V fc ) ® C?^ <8>G k ®Gi . (5.10) Another choice of the energy function W s for the Mooney-Rivlin material is given below (Bathe [1996, p.593]): W s = d (/, - 3) + C 2 (J a - 3) - {Ci + 2C 2 ) ln/ 3 + ^ (m/ 3 ) 2 , (5.1 1) where the parameters C\, C 2 , A have the same meaning as in (5.1). 16 15 By differentiating the identity g^g jq = St with respect to g kl , and by recognizing that = j 9g kt 2 (9jrg g s + 9js9qr), one arrives at (5.10) after postmultiplying the resulting equation with g qp and reusing the identity g jq g vp = 5? once more. 16 Both (5.1) and (5.1 1) are modified forms of the original Mooney-Rivlin model (Truesdell and Noll [1992, p.350]), which corresponds to the first two terms in (5.1 1). 182 The second Piola-Kirchhoff stress tensor S corresponding to W s in (5.1 1) is then given by _ dW, 3 =2 (Cj + dh) 1 - C 2 C - (d + 2C 2 - Aln/ 3 ) C- 1 , (5.12) dE while the material moduli tensor C takes the form C = ~ = 4C 2 1 ® 1 - 4C 2 I - 4 (Ci + 2C 2 - Aln/ 3 ) -xpr + 4AC -1 <g> C" 1 . (5.13) Our numerical experiments show that both models in (5.1) and (5.11) lead to the same results if the same value of penalty parameter A is used. Remark 5.1. To satisfy the incompressibility constraint (i.e., h = 1), the penalty parameter A must be large enough so that the error on incompressibility is negligible (i.e., I 3 is approximately equal to 1). On the other hand, the penalty parameter A cannot be so large that numerical ill-conditioning occurs, and makes Newton's method difficult to converge. Alternatively, the augmented Lagrangian method is designed to avoid the char- acteristic ill-conditioning of the above penalty method, and enforces the incompressibility accurately (Simo and Taylor [1991]), but destroys the desired quadratic rate of asymptotic convergence of Newton's method (See Section 5.4 for some examples). I 5.2.2. The Hyperelastoplastic Model Up to the beginning of the 1980s, computational methods for finite-strain elastoplas- ticity typically relied on hypoelastic extensions of the classical infinitesimal model (Simo and Hughes [1998]), and hence not suitable for applications involving large elastic strains (e.g., metal forging). In the last 15 years, computational approaches based on the multi- plicative decomposition have received considerable attention in the literature. Simo and Ortiz [1985] and Simo [1988a] proposed a computational approach entirely based on the multiplicative decomposition, and pointed out the role of the intermediate configuration in a definition of the trial state via the hyperelastic stress-strain relations. Subsequently, 183 Eterovich and Bathe [1990] and Weber and Anand [1990] used the multiplicative decompo- sition in conjunction with a logarithmic stored energy function and an exponential approx- imation to the flow rule cast in terms of the full plastic deformation gradient. Based on the above multiplicative decomposition, Simo [1992] showed that the closest-point-projection algorithm of infinitesimal plasticity could be carried over to the finite-deformation context without modification. A computational treatment of plasticity is to interpret the local evo- lution equations describing the plastic flow in the framework of the principle of maximum dissipation, which lead to the return mapping algorithm with an operator split procedure. Within a typical time step, an elastic trial state is first computed for prescribed strain in- crements and converged internal variables. Then, in the corrector stage the actual stress is obtained by the closest-point projection of the trial stress state onto the elastic domain. This projection is computed locally at each quadrature point of a typical finite element, and depends exclusively on the functional form adopted by the yield criterion in stress space. For the J 2 flow theory, the closest-point projection reduces to the classical radial return mapping. There are two methods to implement the hyperelastoplastic model at finite strains: (i) multiplicative decomposition of the deformation gradient F (Simo [1988a] and Simo [1988Z?]), and (ii) the spectral form of the right Cauchy-Green tensor C (Ibrahimbegovic [1994], Miehe [1998a], Betsch and Stein [1999]). Our numerical experiments show that while both methods lead to the same computed results, the second method does not require the expensive polar decomposition. For method (ii), we provide a more convenient way to decompose the right Cauchy-Green tensor C. 5.2.2.1. Multiplicative decomposition of the deformation gradient F As already discussed in previous section, the present locking-free solid-shell element formulations are based on the modified Green-Lagrangian strain E. Therefore, a modified deformation gradient F, which is consistent to the modified strain E, is required for the algorithm of the finite strain elastoplasticity (e.g., Simo [1988&]). The evaluation of the modified deformation gradient F consistent with the modified strain E, however, is rather 184 time-consuming. The compatible deformation gradient F c from the displacement field is split into an orthogonal rotation tensor R and a right-stretch tensor U c as follows F C = RU C , (5.14) substituting (5.14) into the compatible Green-Lagrangian strain E c of (3.12) E c = l - (F cT F c - l) , (5.15) the orthogonal rotation tensor R drops out and the compatible stain E c will depend solely on the right-stretch tensor U c as E c = I [U CT (R T R) U c - l] = l - (U CT U C - l) , (5.16) where U c can be obtained by using a polar decomposition on (5.16), and then used in (5.14) for the rotation tensor R by the following R = F C [UT 1 ■ (5.17) The right-stretch tensor U, consistent with the current strain E including the EAS and ANS treatments, can be computed by employing another polar decomposition on the following E = \ (U T U - l) (5.18) With the unmodified orthogonal rotation tensor R calculated from (5.17) and the current right-stretch tensor U obtained from (5.18), the deformation gradient F consistent with the current strain E is computed through the following F = RU. (5.19) Thus, the above procedure to evaluate the deformation gradient F, which is consis- tent to the current strains E, needs twice a time-consuming polar decomposition at each 185 integration point. We refer to Simo and Hughes [1998, p.244] for the closed-form of the polar decomposition. With the deformation gradient F, it is straightforward to implement the return map- ping algorithm for the J 2 finite strain plasticity model. For more details, readers refer to Simo and Armero [1992]. 5.2.3. 1 . Spectral form based on the right Cauchy-Green tensor C Recently, the more advantageous algorithm of isotropic hyperelastoplasticity is based on the principal stretch (Simo [1992], Miehe [1998c]), in which all the algorithms previ- ously developed for the small-strain plasticity can be used directly. The phenomenological description of large-strain elasto-plasticity relies on the local decomposition of the deformation gradient F, F = F e F p , (5.20) where F e is the deformation caused by the stretching and rotation, F p the deformation associated with the plastic flow (see e.g., Maugin [1992, p. 167]). To construct the nonsymmetric tensor Z e associated with elastic stretches, we de- compose the right Cauchy-Green tensor C similar as in (5.20), namely C m Z e C p with Z e := C (C p ) _1 , Z eT := {C p )~ 1 C , (5.21) where C p := F pT F p is the plastic Cauchy-Green tensor. To represent the constitutive relation in terms of the eigenvectors JVj associated with Z e , the standard eigenvalue problem needs to be solved (Z e - A 2 l) Ni = , (5.22) where eigenvalues {A, 2 }- =1 23 represent the elastic principal stretches, and eigenvector Ni and covector AT J satisfies Ni-N s = S{, ij = 1,2,3, (5.23) 186 and N* is normalized with respect to the plastic Cauchy-Green tensor C p N^C^N 1 = 1 , i = l,2,3, (5.24) which leads to simple expressions of C p and C v ~ l as follows C p = j2N i (S>N i , (5.25) i=l 3 C v ~ l = ]T N* $ JV* , (5.26) i=l respectively, where the summation convention is not used in the section. From (5.22), we can express Z e in the following form 3 Z e = '£\?N i ®N i , (5.27) i=l therefore, with (5.25) and (5.27), the spectral form of C becomes 3 C = Z e C p = £ XfNi <g> JV< . (5.28) i=l Assuming the isotropic free energy if) takes the form V> = V(ei,e2,e 3 ,/i) , (5.29) where the logarithmic elastic principal strain e t is defined as the function of elastic principal stretches Aj, namely e< := InA, = hnXf , i = 1, 2, 3 , (5.30) and is the equivalent plastic strain. Define the corresponding principal stress r< as follows r ' := ^' »- 1.2,3. (5.31) According to (5.27) and (5.28), the derivatives of A? with respect to Z e and C are respectively <9A 2 <9A 2 ^^rtQNi, -^L = N i ®N i . (5.32) 187 With (5.30), (5.31) and (5.32) 2 , the stresses S in the spectral representation are = dip dE dc m 2\^—— = 2Vr— — = — iV* ® AT (5 33) The Kuhn-Tucker loading/unloading condition is 7 >0, 0<O, 70(r : ,r 2 ,r 3 ,?/) = O, (5.34) where is the yield function, 7 plastic parameter, and y = ~. The flow rule for the oh internal variables h and C p in spectral form can be expressed as the ordinary differential equations with respect to the time d<j) h = Tjj, (5.35) C p ~ l = -2 1 C*- l "£^N i ®N i . (5.36) i=l Remark 5.2. As mentioned in Simo [1988a], the Kuhn-Tucker optimality conditions and constraints on the evolution equations for the internal variables can only be obtained naturally through the principal of maximum plastic dissipation. Consider the following local dissipation inequality (Simo and Hughes [1998, p.229]) P = \ S -- b -Jt^ C ^^ h ) 2{ b 2 dC ■ dC^ ~dh k -°- (537) Since S=^L = 2— dE dC ' (5.38) and denote y := — , (5.37) is rewritten as oh V-=-g^--C P ~ 1 -yh>0, (5.39) 188 taking the derivatives of eigenvalue problem (5.22) while keeping C fixed, along with (5.2 1) 2 , it becomes (< ' CdC p ~ x - ^h dCP ~ 1 ) N * + ( CCP ~ l ~ \ 21 ) dNi = 0, (5.40) by multiplying JVj at both sides of (5.40), using the orthogonality condition JVj • dNi = 0, and with (5.28), we have dC p ~ 1 it follows that XtNiQNi, (5.41) dip 1 - t&Mx-t&x***- w dC v ~ l £j d\t dC*- 1 fr[dX\ With (5.32) 2 , we have dib A dip dXf JL, 0d> , — = V— — - = V — AT 1 g> A/" 1 (5 43) Therefore, with the above (5.42) and (5.43), and combining the expression of C in (5.28), C p in (5.25), and S in (5.33), we obtain the relation = C^C = l -CSC p . (5.44) dC p ~ l dC 2 With (5.33) and (5.28), CS in (5.44) becomes 3 cs = Yl T i N i® Nl . ( 5 - 45 ) i=l thus the dissipation inequality in (5.39) can be rewritten as follows 1 3 P = --J^TiNi ® APC P : Cf" 1 - yA > . 2 i= i (5.46) The classical rate-independent plasticity model in the spectral form is obtained by postulating the following local principle of maximum dissipation: For admissible (Ti, h), which satisfies the yield criterion (f> (r^, h), the actual state (Ti, h) maximizes the dissipation function V. Using the Lagrange multiplier method, we define the constrained minimization problem in the following C (r,, y, 7) := -V (r<, y) + 7 (r<, y) , (5.47) 189 where 7 is the plastic consistency parameter. By finding the extremal point of C, the con- dC strained minimization problem of (5.47) leads to the flow rule in (5.35) (i.e., — = ) and dy (5.36) (i.e., — = ), and the Kuhn-Tucker loading/unloading condition of (5.34). I OTi For the evolution equation, the plastic hardening variable h is obtained by a standard implicit backward Euler method with the time increment At = t n+ \ — t n on (5.35) d({> K+i = h n + P , dy , = ln+1 At . (5.48) n+l As originally proposed by Weber and Anand [1990], we integrate the flow rule of C p_1 in (5.36) by means of the exponential map. The advantage of this method is two folds: i) preserve the plastic incompressibility condition of the pressure insensitive yield function 0; ii) return mapping algorithm of small strain plasticity can be used without any modification. By using the backward Euler integration with an exponential map on (5.36) (simi- larly, for y = c(t)y, it follows that y n+ i = [exp (c n+ i At)] y n ), it yields C p n ~ + \ = exp (-20 E f^AT ® n] C*T 1 , (5.49) V i=l OT i /„+! where C v ~ l is associated with the eigenvalue problem (5.22) for the trial state, that is (C n+1 CT 1 - A?l) N\ = , (5.50) which gives C n+l Cl~ x = £A?iVj S N u . (5.51) i=l In parallel, C„+\ is associated with the eigenvalue problem at t n+ i (C n+1 C p n -\ - Xfl) N t = , (5.52) which gives Cn+iCHl^^Ni^N*, (5.53) i=l 190 where iVj is the eigenvectors associated with the eigenvalue A* for time £ n+1 . Similar to (5.25) and (5.26), the plastic Cauchy-Green tensor C p n and its inverse at the trial state are expressed as follows 3 3 C P n = E N \ ® N \ . CT 1 = E ^ ® ^ • (5-54) i=l i=l Substituting (5.54) 2 into (5.51), and recalling (5.28), we have 3 3 C n+1 = E AfiV* ® N* = E A i ^ « ^ • (5-55) Therefore, the eigenvectors N t at t n+x and jV{ at the trial state are proportional \t Ni = ^-N\ , (5.56) by multiplying AP and JV" at both sides of (5.56), and with (5.23), the eigenvectors N l associated with A t at t n+l and AT* 1 associated with X\ at the trial state are also proportional N l = ^ t N u . (5.57) Considering the above (5.57) in (5.54) 2 , we have the following 3 3 \t2 CT 1 =E^® = E T2 N ' ® N * . (5-58) i=l i=l A z substituting (5.58) into (5.49) and using (5.26), it becomes \ 2 - ex P (-2/?f^) A? , (5.59) with the definition of (5.30), the update of the principal strain e* is by simply taking the logarithm on both side of (5.59), namely « (5-60) Denote {e t }, ^-^},-r-{ ? }-|^| l i = 1,2,3, 191 d(j> d(j> dtp or Oy Oh _ d 2 j) _ dr_ ^ _ d 2 ip _ dy_ dede de ' h dhdh dh ' d 2 (f> dn d 2 (j) dn h = a a = "H - ' -^h = a r, — i p.Ol) or or Or oyoy oy the solution of the set of nonlinear algebraic equations (5.48), (5.60) along with the yield criterion cf> = is typically obtained by a Newton procedure. A Newton procedure based on the systematic linearization of these equations gives rise of a plastic-corrector return to the yield surface based on the concept of closest-point projection. We write the plastic updates in (5.48), (5.60) and yield condition (ft = at time in the form of r t = e - e l + 0n = , (5.62) r h = h - h l + Pn h = , (5.63) (f>(r,y) = 0, (5.64) linearizing (5.62)-(5.64) gives the incremental form at iteration (k) of time t n+ y, r e + Ae + A(3n + pAn = , (5.65) r h + Ah + Apn h + pAn h = , (5.66) 4> + nAr + n h Ay = . (5.67) By making use of the following from (5.61) At = £Ae , Ay = S h Ah , (5.68) An = TAr , An h = F h Ay , (5.69) and substituting (5.68) into (5.67), substituting (5.69) into (5.65) and (5.66), it yields r e + £~ l SAe + Af3n = , (5.70) 192 r h + S^£ h Ah + Af3n h = , (5.71) <t> + n£Ae + n h £ h Ah = , (5.72) with I = [E-'+p^y 1 , 4 = fa 1 +pJ r h)~ 1 , (5.73) then we can solve for the increments A/3, Ae, Ah from (5.70)-(5.72) as follows A/3 = (</> - n T S r e - n h S h r h ) , (5.74) Ae = -5" 1 5(r f + A(3n) , (5.75) Ah = S^S h ( r h + A(3n h ) , (5.76) with D = n T £n + E h n 2 h . (5.77) With the above obtained increments, we update the strains, internal variables and plasticity parameter for the iteration [k + 1) at time t n+ i. The local Newton iteration procedure is continued until the convergence to the yield surface within a sufficient tolerance. The Newton's method relies on the fourth-order consistent tangent moduli C (also called the algorithmic tangent moduli) , which is crucial in the development of the material tangent stiffness matrix (see Section 3.3). In Simo and Taylor [1985], it was first shown that the disappointing rates of convergence exhibited by Newton-type iterative methods arise from lack of consistency between the continuum elastoplastic moduli and the return mapping algorithm. The development of the consistent tangent moduli is based on a sys- tematic linearization of the stress update algorithm, which is defined as where S at t n+ i is obtained from (5.33). Since the principal stress T t at time t n+ i is the function of t\, with (5.32) 2 and (5.33), the derivation of (5.78) leads to the algorithmic tangent moduli C (see also Ogden [1984], Zienkiewicz and Taylor [2000b, p. 343] for the details) in the following ^ . Ti N li ® N ti c = Y^2 d ^ i J^ ) N ti ®N ti + Y. 2 t-i dC ' £"A? dC 193 N u <g> N u ® N tj ® AT' J i=l j=lj#i where the elastoplastic moduli is defined in the eigenspace as follows which can be derived based on the linearization of (5.60), (5.48) along with = 0. The elastoplastic moduli £*? is derived in Appendix A.5, and the term g tJ listed in (A.48). Since we develop the present solid shell formulation in the convected basis, it is nec- essary to transform the second Piola-Kirchhoff stress tensor S and the consistent tangent moduli tensor C from the basis of eigenvector N u to the basis G u namely S = S%N tl ®N t] = S ij Gi®G 3 C = C? jkl N u ®N tj ®N tk ®N tl =C ljkl G i ®G j ®G k ®G l , (5.81) which involve the transformation procedure similar to that used between the Cartesian basis e l and the convected basis G{ in Section 3.3. In summary, we identify {C, C p , h} as the state variables. Once the state variables are known, the stress tensor S and the consistent tangent moduli C will be determined through (5.33) and (5.79) respectively. The detailed implementation on the stress update and consistent tangent moduli are listed in Appendix A.4 and Appendix A.6, respectively. 5.3. Explicit Time Integration Method for Solid-Shell Elements Here we discuss the conditional-stable explicit method associated with the present solid-shell element, in which the reliable full numerical integration is employed. Since the maximum stable time-step size At decreases with the mesh refinement (see, e.g., Be- lytschko, Liu and Moran [2000, p.314], LS-DYNA [1998, Chap. 19]), it is beneficial to use the present element, which has coarse mesh accuracy, and needs only one element through the thickness, in explicit analyses of shell structures. Reduced-integration (RI) elements 194 have been widely used together with explicit time integration (DYNA3D [1993], ABAQUS [200 1 ]). Due to the use of ad hoc assumptions on the kinematics and the material properties, the proper stabilization (hourglass control) techniques for the spurious zero-energy modes caused by the RI scheme are, however, still an active research area, especially for physi- cally nonlinear problems such as crashworthiness problems (Zhu and Cescotto [1 996], Zhu and Zacharia [1996], Zeng and Combescure [1998]). Furthermore, reduced-integration elements are highly sensitive to mesh distortion (see, e.g., Stanley [1985]). Although there is a disadvantage with the need for using small time increments for stability, the advantages of explicit methods are significant in that the construction, stor- age, decomposition and back substitution of the effective tangent stiffness matrix, which is required in implicit methods, are completely avoided. There are many situations where explicit methods are preferable. For example, in high-speed events such as car crash simu- lations, a small time step is required due to the noise introduced by the contact and impact between different structural parts. Two types of mass matrices can be considered: The non-diagonal consistent mass matrix and the diagonal lumped mass matrix. To increase computational efficiency, lumped mass matrix is often chosen to avoid the decomposition and back-substitution. Although the procedures for diagonalizing the mass matrix are quite ad hoc and questionable, espe- cially for high-order elements and stress-resultant shell elements (Zienkiewicz and Taylor [1989, p.605], Hughes [1987, p.565]), the lumped element mass matrix for the present eight-node solid-shell element can be obtained easily without problem. One common ap- proach is the row-sum technique (Zienkiewicz and Taylor [1989, p.474]), in which the diagonal entries of the lumped element mass matrix raU are obtained by ndof m \i = E K . (5.82) j=i where the sum is over the entire row of the consistent element mass matrix [m c 1 e ^ndofxndcrf where ndof is the number of degrees of freedom in an element (see Vu-Quoc and Tan [20026]). 195 Alternatively, the lumped element mass matrix from the following expression rrii = J NpdV , n(«> can also be evaluated directly (5.83) where in the element domain Bq \ N is the matrix containing the shape functions corre- sponding to degrees of freedom of the element nodes, and p the mass density. Both (5.82) and (5.83) lead to the same lumped element mass matrix for the present element. The global semi-discretized nonlinear ordinary differential equation (ODE) is in the form Mu = F ext - F int , (5.84) where M is the assembled mass matrix, and the internal force F is computed as follows -tint nel / , . ~ \ J stiff K ua ^aa J stiff ^ua \ r "aa\ J EAS ) ' t(e) (5.85) with the matrices fc^, k^ a , fg\ s being the terms associated with the EAS method in (3.78), and EAS parameters are updated through (3.79). To solve the nonlinear ODE (5.84), the effective and most widely used explicit method is the central difference method. In a typical time step [£ n ,£ n+ i] with time-step size t n+ i — t n = A£, the displacement u n+i at t n+ i is updated by using the displacement u n , the velocity u n , and the acceleration u n at t n as follows At 2 u n+1 = u n + u n At + u n —— By using (5.84) and (5.86), the acceleration u n+1 at time t n+ i is obtained via u n+l = M- 1 F ext (vi) - F™ (u n+1 ) (5.86) (5.87) where only divisions are required for the solution of u n+ i due to the diagonal form of M, in contrast to the expensive decomposition and back-substitution needed in implicit 196 methods such as the Newmark and energy-momentum conserving algorithms (Vu-Quoc and Tan [20026]). With u n+l from (5.87), the velocity u n+1 at t n +i is approximated by w n+ i = u n + -y (u n + u n+ i) . (5.88) Updating for the displacement u n+l by (5.86) does not require the solution of any algebraic equations. Thus, in this sense, explicit integration is simpler than implicit inte- gration. As shown in Appendix A.3, the explicit program is a straightforward evaluation of the governing equations and the time integration formulas. It can be seen in (5.85) that the computation of the internal nodal forces involves the calculation of strains, stresses, and the constitutive matrix. When the element nodal forces are calculated, they are assembled to the global array according to the node connectivity. By prescribing the nodal velocities at prescribed velocity boundaries (see, e.g., in Appendix A.3), the correct nodal displace- ments result from (5.86). The reaction forces at prescribed velocity nodes can be obtained from the total nodal forces Rn +l = F& - FJJ, • (5.89) Since the time step in explicit integration must be below a critical value (otherwise the numerical solution will blow up), it is not recommended to use explicit methods in quasi-static or low-speed events such as the springback effect in metal forming process. On the other hand, it is appropriate to combine explicit methods and implicit methods to maximize computational efficiency, that is, for example, in metal forming or crash analysis, explicit time integration can be used for the initial time stepping and then a static or implicit dynamic solution be used for the rest (see, e.g., ADINA [2002]). 5.4. Numerical Examples The finite element formulation of the present low-order solid-shell element for non- linear analysis of shell structure presented in the previous sections has been implemented in both the Finite Element Analysis Program (FEAP), developed by R.L. Taylor (citef :zie.89a), 197 and the NIKE3D by the Lawrence Livermore National Laboratory (NIKE3D [1995]), and run on a Compaq Alpha workstation with UNIX OSF1 V5.0 910 operating system. In each element, the mass matrix is evaluated using the Gauss integration 2 x 2 x 2, the tangent stiffness matrix, the dynamic residual force vector are evaluated using the Gauss integration 2 x 2 in the in-plane direction, three Gauss points in the thickness direction for Mooney- Rivlin material, and five Gauss points in the thickness direction for elastoplastic material. Below we present the examples involving the static and dynamic large deformation analysis with nonlinear materials. In the nonlinear dynamics analysis, we use the trapezoidal rule for implicit time integration (with the consistent mass matrix) and the central difference method for explicit time integration (with the lumped mass matrix), without introducing numerical dissipation. 5.4. 1 . Large Deformation of Rubber Shells Here we analyze the large deformation of rubber shells by using the present solid- shell element and Mooney-Rivlin material model. For the augmented Lagrangian method, a function of 7lnJ 3 is appended to the strain energy function W s in either (5.1) or (5.11), where 7 is augmented Lagrangian multiplier. The augmented Lagrangian procedure can be accomplished by a nested iteration algorithm (Zienkiewicz and Taylor [2000a, (p.323)]). Within a typical iteration of one step, we solve the linear system with fixed penalty pa- rameter A. Then the Lagrangian multiplier 7 is updated by 7 = 7 + Am/ 3 at the end of nested iteration . The nested iteration can be repeated with the increased penalty parameter to reach the desired accuracy for the incompressibility constraint. Here we use an initial value of A = 250 for the penalty parameter. Our numerical experience on the uniform extension/compression test of one element (Bathe [1996, p.593]) showed that with four nested iterations the incompressibility can be enforced to the tolerance of 10 -8 in the vol- ume change between the initial configuration and the deformed configuration, compared to 10~ 5 by using the penalty method with a large value of penalty parameter A = 5000. 198 5.4. 1.1. Stretch of a rubber sheet with a hole This problem has been analyzed by Gruttmann and Taylor [1992]. The material con- stants for Mooney-Rivlin model are C\ = 25 and C 2 = 7. The length of the square is L = 20, the radius of the circle is R = 3, and the thickness h = 1 (left of Figure 5.2). Due to the symmetry, only one quarter of the sheet has been modeled with 64 solid-shell ele- ments. The augmented Lagrangian method is used here, with five nested iterations for each load step. Figure 5.2 depicts the initial geometry and the corresponding final deformed mesh configuration for q = q = 90. A full agreement with the membrane element devel- oped by Gruttmann and Taylor [1992] is shown in left of Figure 5.3. It is clear that the large strains and the thickness stretching are involved in the present problem. For instance, the sheet thickness at point D becomes one half of the initial thickness. It is interesting to note that the sheet thickness at point B is increased rather than decreased (right of Figure 5.3). Figure 5.2. Stretch of a rubber sheet with a hole: Initial configuration (left) and deformed shape at q Q = 90 (right). 5.4. 1.2. The snap-through of a conic shell Next we show the robustness of present solid-shell elements with large elastic strains. This problem was appeared in Li, Hao and Liu [2000] for the application of their meshless method, in which 12,300 particles with three particles in the thickness are used. A total of 1,800 solid-shell elements (3,720 nodes) are used in the discretization, with only one element in the thickness direction. It is noted that the ongoing intensive research is directed 199 Figure 5.3. Stretch of a rubber sheet with a hole: Load-displacement diagram (left) and thickness stretching (right). to make meshless methods more computationally efficient, which includes the interpolation scheme, numerical integration procedures and techniques of imposing boundary conditions (De and Bathe [20016], De and Bathe [2001a], Atluri and Shen [2002]). The material and geometric properties are Ci = 18.35, C 2 = 1.468, A = 1.468 x 10 3 , p = 1.4089 x 10" 4 , Rtop = 1, Rbot = 2, H - 1, h = 0.05 , (5.90) where C\, C 2 are the material constants, A the penalty for incompressibility, p the initial density of the rubber shell, and R top , R bot the radius at the top and the bottom of conic shell, respectively, H the height, and h the thickness of the conic shell. The time increment is chosen as At = 25 x 10 _6 sec, and total 400 steps are used in the implicit Newmark method. In the computation, we fix the bottom edge of the conic shell and do not allow the horizontal movement of the upper inner edge, and prescribe the vertical displacement of upper inner edge such that it drags the whole shell structure down. At the end of the computation, the conic shell turns inside out. In Figure 5.4-5.5, several snap-shots are taken to form a deformation sequence. The deformation is large and involves both material nonlinearity and geometry nonlinearity. Such deformation process 200 belongs to a so-called snap-through instability problem because the reaction force at top edge is alternating along with the advance of vertical displacement of the top edge (see Figure 5.6). In the static case, the magnitude of the reaction force is much smaller than in the dynamic case, which is true when the inertial effect is not considered (left of Figure 5.6). '^7777777777:' Figure 5.4. Snap-through of the conic shell: Initial configuration (left) and deformed shape at t = 2.5 x 10~ 3 sec (right). Figure 5.5. Snap-through of the conic shell: Deformed shape at time t = 5.0 x 10 3 sec (left) and t = 10.0 x 10~ 3 sec (right). Finally, we point out that, both the penalty method and augmented Lagrangian method leads to similar behavior (right of Figure 5.6) in this problem, while the latter needs more iterations at each time step (Table 5.1). 5.4. 1.3. Large motion of the pinched cylindrical shell This problem was also appeared in Li et al. [2000] for the application of their mesh- less method. We prescribe the inward radial displacement for two opposite nodes of the inner-surface at the middle section of the cylinder (Figure 5.7). Disp. control 201 k sialic | | u Li 1 iji 1 n Figure 5.6. Snap-through of the conic shell: Deflection versus reaction force curve by penalty method (left) and augmented Lagrangian method (right). Table 5.1. Snap-through of the conic shell: Convergence results for both penalty method and augmented Lagrangian method at a typical time step (energy norm) Iter. Penalty 1 st augm. 2nd augm. 3rd augm. 4th augm. 5th augm. 3.20E-02 1.44£-03 1.32£-05 2.64£-06 4.74£-07 8.30£-08 1 7.19£-03 7.35£-03 2.34E-08 1.20£-10 8.30£-13 8.18E-15 2 4.86£-05 2.80£-06 2.48£-16 1.97E-20 2.86£-24 5.41.B-28 3 1. 16£-08 2.10J5-12 4 9.02E-16 3.02£-23 The material properties are the same as in (5.90) and geometric properties are R = l, H=l, h = 0.02, (5.91) where R is the radius, H the height, and h the thickness of the cylinder. Because of the symmetry, only one-eighth of the cylinder needs to be modeled by 32 x 32 x 1 solid-shell elements with 2,178 nodes (Figure 5.7). For the temporal integration of explicit central difference algorithm, the time increment is At — 0.5 x 10 _6 sec and total 20,000 time step has been taken to finish the run. For the integration by implicit Newmark algorithm, the time increment is At = 10 x 10 -6 sec and total 1,000 time step has been taken. In Li et al. [2000], total 30,300 particles with three particles in the thickness are used and 21,000 time steps are taken in the explicit analysis. Both the implicit method and the explicit method lead to the similar deformation sequence shown in Figure 5.8- 5.9, The deformation of cylinder under pinched loading is drastic. At the end of our computation, the two opposite points of inner surface of the 202 cylindrical shell come together. Compared to Li et al. [2000], the deformation from the current calculation is more severe and appears more buckling modes. , FE model H Figure 5.7. Large motion of pinched cylindrical shell: Geometry and loading. Figure 5.8. Pinched cylinder: The deformation at time t = 2.0 x 10 3 sec (left) and £ = 4.0 x 10~ 3 sec (right). The relation between the reaction force and the deflection both at the pinch point is presented in Figure 5.10. It is observed that for both the implicit method and the ex- plicit method, the results of reaction forces are very close, which justify the use of explicit method in the present solid-shell element for the high-speed dynamics. Figure 5.9. Pinched cylinder: The deformation at time t = 8.0 x 10 3 sec (left) and t = 10.0 x 10 _3 sec (right). 5.4.1.4. Rubber hemispherical shell The clamped rubber hemispherical shell is subjected to a point load at the pole. This problem is used to demonstrate the localized effects of thickness change and importance of applying the external surface loading for moderately thick shell problem. Without any modification on the element formulation, the surface loading can be considered naturally in the present solid-shell element. The material and geometric properties are C, = 25, C 2 = 7 , A = 1. x 10 3 , R = 26.3, h = 4.4 , (5.92) where C\, C 2 are the material constants, A the penalty for incompressibility, respectively, and R the radius, and h the thickness of the spherical shell. Due to the symmetry, one quadrant is modeled by 192 solid-shell elements (Figure 5.11). Displacement control is used to drive the top of the hemisphere down. Two different loading cases, top surface loading at point A and both surfaces loading at point A and B to approximate the midsurface loading in classic shell element (with inextensible director), are considered. The both-surface loading keeps the thickness unchanged between point A and B, and 204 0.035 1 1 1 1 1 1 r -0 005 1 1 1 1 1 1 1 1 1 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 It Figure 5.10. Large motion of the pinched cylindrical shell: Deflection versus reaction force curve. produces a "kink" at point B, which is true only for thin shells (left of Figure 5.12). The top- surface loading exhibits the anticipated localized effects and the thickness change by about 70% between point A and B, which produces the physically more reasonable deformation (right of Figure 5.12). The relation between the deflection at loading location and the corresponding total force to produce such deflection for both cases is presented in Figure 5.13. The difference on two loading cases shows the importance of localized thickness change effects. 5.4.2. Large Deformation of Elastoplastic Shells For finite deformation J 2 plasticity, the free energy function is assumed to take the uncoupled form as follows 1> (ei, e 2) es,h) = ^A p j + fij^ (e,) 2 + hih 2 , (5.93) where A and \i are the Lame parameters, H is a parameter which characterizes the isotropic hardening in the material, and h the equivalent plastic strain. The Von-Mises yield criterion 205 Clamped Figure 5.1 1. Rubber hemispherical shell: Mesh, boundary conditions and loading for the one quarter of shell. Figure 5.12. Rubber hemispherical shell: Displacement loading at both surface shows "kink" at point B (left), and displacement loading at top surface has the smoothness at point B. The thickness between point A and B is changed by approximately 70% (right). is expressed in terms of principal stresses as <f>(T U T 2 ,T 3 ,y) = 1 EW- [En t=l " \i=l where r y is the initial flow stress, y = Hh. {t v + v) (5.94) 206 450 — top surface loading — both surface loading 400 2 4 6 W 8 10 12 Figure 5.13. Rubber hemispherical shell: Load-deflection curve showing the difference of the two loading cases. 5.4.2. 1 . Bending of a cantilever beam This example investigates an elastoplastic cantilever beam illustrated in Figure 5.14 for several aspect ratios L/h, and confirms the correctness of the current solid-shell element for elastoplastic material. The mesh for the plate is 20 x 1 x 1. The material properties are where E, [i, r y , H are the Young's modulus, the Poisson's ratio, the initial yield stress, and the isotropic hardening parameter, respectively. Numerical results are compared to Eberlein and Wriggers [1999] where the solutions are computed from the classical 5-parameter shell element (volumetric locking does not appear in elements with the plane-stress condition (Hughes [1987, p. 193])). The good agreement in all range of aspect ratios can be observed. For the aspect ratio of L/h = 100, the final deformed configuration with the contour of the equivalent plastic strains at the £=1.2xl0 7 , /x = 0.3, T y = 2.4 x 10 4 , H = 1.2 x 10 5 , (5.95) 207 plate surface is shown in right of Figure 5.14. clamped plastic stra 0.0047 0,0039 0026 0.0013 0003 a 0.0000 Figure 5.14. Out-of-plane bending: Geometry and mesh of cantilever plate (left), and deformed shape and upper-surface contour plot of equivalent plastic strains (right). A load deflection curves for different aspect ratio are shown in Figure 5.15-5.16. Compared to our seven-parameter EAS formulation, the solid-shell element with five- parameter EAS in Miehe [1998&] and Klinkel et al. [1999] behave stiffer and cannot obtain the collapse load for the thick and moderate thin plate (Figure 5.15), which may be caused by the inability of the element to deal with the volumetric locking in the plastic deforma- tion stage. It is observed for the thin case (aspect ratio = 1000) the plastic stage doesn't appear till the end of loading, so two Gaussian points through the thickness are enough for this elastic problem, and the results from Miehe [1998&] and Klinkel et al. [1999] is close to the present though their element formulations cannot pass the out-of-plane bend- ing patch test (Subsection 3.5.1). The plasticization appears at load F = 475 for aspect ratio L/h = 10, and F = 5 for aspect ratio L/h = 100. Cantilever beam loaded by shear lores, L/h • 1 Cant it eve t beam loaded by shear tores, L/h « 100 pre sent solid-shell O Ebenem etal. [1999] — 5-param L AS 10 Figure 5.15. Out-of-plane bending: Load deflection curve for aspect ratio L/h — 10 (left) and for aspect ratio L/h = 100 (right). 1 2 3 4 5 6 7 W Figure 5.16. Out-of-plane bending: Load deflection curve for aspect ratio L/h = 1000. Since the exact tangent (Jacobian) matrix is employed in the Newton's solution pro- cedure, the quadratic rate of asymptotic convergence was actually observed in all problems we examined. Table 5.2, which depicts the values of the Euclidean norm of both the resid- ual and the energy norm at each iteration, clearly exhibits the quadratic rate of asymptotic convergence, which is in the sharp contrast to the extremely slow convergence (hundreds of iterations in one step is not uncommon) with the hypoelastoplastic rate-form material model (Choudhry and Wertheimer [1997]). 5A2.2. Elastoplastic response of a channel beam A far larger and equally important class of structures are the nonsmooth shell which consists of folds, kinks or branches. Here we present the warping of an angle iron to prove the validity and applicability of the proposed solid-shell element formulation, in which the simulation of such structures on both the overall behavior and local stress concentration ef- fect at one time becomes straightforward, and does not introduce any assumption on the in- 209 Table 5.2. Out-of-plane bending: Convergence results for plate with L/h = 100 (residual norm, energy norm) Iter. Step 1 (F=0.5) Step 10(F=5.0) Step 13 (F=6.5) 2.505-01,8.275-02 6.835+00, 7.735-02 1.765+01,7.275-02 1 1.005+04, 1.795+00 9.025+03, 1.455+00 8.215+03,1.205+00 2 3.965+00, 3.295-07 5.275+00,3.625-04 4.585+01, 1.53S-02 3 3.465-05,3.925-14 3.83E+01, 2.59£7-05 2.125+03,8.335-02 4 5.515-09,7.105-25 4.145-03,9.195-10 8.225+01,1.115-02 5 1.305-04,3.015-16 3.835+03,2.575-01 6 1.705-08,7.015-24 2.515+00,3.415-03 7 1.995+03,6.915-02 8 1.345-01,5.705-06 9 3.405+00,2.025-07 10 6.835-06,6.165-14 11 4.575-08,3.995-23 terconnection or penalty parameters for the shell intersection. A cantilever beam subjected to a point load is considered as given in Figure 5.17. Because the centroid (inside of the open cross section) and the shear center (outside of the open cross section) of the beam do not coincide, if the concentrated forces are not applied at the shear center of the cross sec- tion, the beam section will twist significantly. Originally, the purely elastic material behav- ior of this example was presented by Chroscielewski, Makowski and Stumpf [1992], where the numerical results show a strong dependence on a penalty multiplier which accounts for the drilling stiffness within their shell formulation. The same example for elastoplasticity was also investigated by Eberlein and Wriggers [1999], where the converged results are presented for their 6/7 -parameter shell concept along with a penalty method for the shell intersection. Since the present element uses the upper and lower surface to describe the deformation, the intersection can be simply modeled by solid elements. For the detailed analysis of the intersection, a refined solid element mesh can be used, which can be easily connected to other shell parts modeled by solid-shell elements. It is noted that it is not possible for stress-resultant shell element to describe the detailed strain and stress distribu- tion at the intersection (Chroscielewski et al. [1997]), while the solid-shell element offers 210 a more realistic and convenient representation for the physical structure. The ideal elastoplastic material properties are used as follows E = 10 7 , n m 0.333, T y = 5.0 x 10 3 , H = Q, (5.96) where E, fi, r y , H are the Young's modulus, the Poisson's ratio, the initial yield stress, and the isotropic hardening parameter, respectively. For the computations, a discretization of 1,584 elements is used as shown in Fig- ure 5.17. In Figure 5.20, the load-deflection curves for both the elastic material and elasto- plastic material obtained from the present element coincide very well with solutions by Betsch et al. [1996] and Eberlein and Wriggers [1999], respectively. The deformed con- figurations for u = 4 are illustrated in left of Figure 5.18 for the elastic solution and right of Figure 5.18 for the elastoplastic solution. For the elastic solution, the buckling of the upper flange can be observed in the vicinity of the clamped edge, whereas the free end is twisted, since the external load does not act in the shear center. The elastoplastic solution also shows a buckling phenomenon in the upper flange, but it is more concentrated than in the purely elastic case. Since the plastic strains occur very early during the deformation process as shown in Figure 5.20, the equivalent plastic strains for the deformed configura- tion u = 0.2 are reported in left of Figure 5.19. As expected from the beam theory, there is a maximum of plastic deformation in the lower and upper flange at the clamped edge of the steel channel. The equivalent plastic strains for the deformed configuration u = 4.0 are shown in right of Figure 5.19. It is noted that for the current shell with intersection, there are two ways to model the intersection as in Mesh 1 and Mesh 2 of Figure 5.21. From our numerical experience, the results from the Mesh 2 is stiffer than Mesh 1, which is more pronounced for the linear elastic material (Figure 5.20). The possible reason for it lies on the sensitivity of assumed strain method to the mesh distortion through the thickness. For folded shells with multi- intersections, Mesh 3 in Figure 5.21 is more advantageous for the modeling, compared to the more involved shell elements with drilling couples (Chroscielewski et al. [1997]). Figure 5.18. Channel beam: Deformed mesh for elastic material at u — 4.0 (left) and deformed shape for elastoplastic material at u — 4.0 (right). 5.4.2.3. Pinched hemisphere The present example investigates an elastoplastic hemisphere, which has been calcu- lated previously by Simo and Kennedy [1992] by means of a 5-parameter shell formulation in combination with a plasticity model formulated in stress resultants, based on an addi- 212 Figure 5.19. Channel beam: Deformed mesh and contour plot of equivalent plastic strains at u = 0.2 (left) and at u = 4.0. (right). - present elastic model *Ncoi net eiem - elasioptasix: model nl comer etem U U Figure 5.20. Channel beam: Load deflection curve by model with solid elements at corner (left), and model without solid elements at corner (right). tive decomposition of the strains into elastic and plastic parts. The radius and thickness of hemisphere are R = 10 and h = 0.5, respectively. Figure 5.22 illustrated the mesh and the boundary condition for the present calculation, where total 432 solid-shell elements are used to model the one quadrant of the hemisphere due to the symmetry. The material properties are E = 10, n = 0.2, r y = 0.2, H = 9, (5.97) where E, fx, T y , H are the Young's modulus, the Poisson ratio, the initial yield stress, and the isotropic hardening parameter, respectively. Figure 5.23 shows the deformed hemisphere including an outer-surface contour plot of the equivalent plastic strains at a load level of F = 30 x 10~ 3 . The load-displacement curves are depicted in Figure 5.24, where the results reported in Eberlein and Wriggers [1999] are included. Our results agree with Eberlein and Wriggers [1999] very well. 213 "71 Mesh 1 Mesh 2 Mesh 3 Figure 5.21. Mesh at the cross section of shells with intersection. Figure 5.22. Pinched hemisphere: Geometry and mesh of one quadrant of hemisphere. 5.4.2.4. Elastoplastic response of a simply supported plate We consider the elastoplastic deformation of a simply supported square plate. The square plate has a length of L = 0.508 and thickness h = 2.54 x 10" 3 . Due to the symmetry of geometry and the boundary conditions, only a quarter of the plate has been discretized with 16x16x1 solid-shell elements (Figure 5.25). It is simply supported so that horizontal displacements and rotations may occur. Only the transversal displacements are set to zero 214 plastic stra 0.0486 0.0347 0.0208 0.0069 0.0041 0.0020 0.0000 Figure 5.23. Pinched hemisphere: Deformed shape and outer-surface contour plot of equiv- alent plastic strains. Figure 5.24. Pinched hemisphere: Load F versus the deflection under load F. at the boundaries of the upper surface of plate. The ideal elastoplastic material properties are E = 6.9 x 10 10 , n = 0.3, T y = 2.48 x 10 8 , H = , (5.98) Figure 5.25. Inflation of a plate: Mesh, boundary conditions and loadings for the one quadrant of square plate. where E, fi, T v , H are the Young's modulus, the Poisson ratio, the initial yield stress, and the isotropic hardening parameter, respectively. A uniform transverse dead-load of p = 10 4 has been applied at the upper surface of plate. Figure 5.27 depicts the load deflection curve where the load factor / in p — fp has been plotted as a function of the vertical displacement w of the center point of the square plate. The curve of the present formulation is in agreement with the converged results from the high-order shell element in Schieck, Smolenski and Stumpf [1999], which is available up to / = 27. Figure 5.26 displays the deflection of the plate at the load level / = 70, along with the upper-surface contour plot of the equivalent plastic strains. The remarkably distorted element in the corner zones of the plate is observed. And the localization of the plastic strains in the four corner zones of the square plate is appeared. 5A2.5. Elastoplastic response of a pinched cylinder The example is concerned with the elastoplastic deformation of a thin-walled cylin- der with large equivalent plastic strain (about 100%). This example has already been inves- 216 plastic stra 0.2567 0.2007 0.1345 0.0701 0.0141 0.0005 Figure 5.26. Inflation of a plate: Deformed shape and upper-surface contour plot of equiv- alent plastic strains. 0.16 Figure 5.27. Inflation of a plate: Load factor / versus vertical displacement w of center point of square plate. tigated by Simo and Kennedy [1992], Wriggers, Eberlein and Reese [1996], and Eberlein and Wriggers [1999]. The geometry and the boundary conditions of the cylinder are dis- played in Figure 5.28, where the mesh of 32 x 32 x 1 solid-shell elements are used for one 217 eighth of the cylinder due to the symmetry. The cylinder is loaded with two radial pinched displacements in the middle of the structure. At both ends the boundary conditions have been prescribed such that the circular shape of the end cross section is preserved and free deformation in axial direction is allowed. The von Mises-type elastoplastic material with isotropic hardening response is governed by the material parameters £ = 3000, M = 0.3, 7, = 24.3, # = 300, (5.99) where E, (M, r y , H are the Young's modulus, the Poisson's ratio, the initial yield stress, and the isotropic hardening parameter, respectively. FE mesh 32 x 32 x 1 Rigid diaphragm L = 600 R = 300 h = 3 H *T — 1 1 • • 1 1 • 1 1 1 ■ 1 1 ■ 1 . 1 • 1 ' ' . 1 • / « /.* / » w w Figure 5.28. Pinched cylinder: Geometry and loading. Figure 5.29-5.30 display the deformation shapes of half of the cylinder at w = 150, = 200, w = 250, and w = 280, along with the the contour plot of the equivalent plastic strains on the outer surface of the shell, respectively. The maximum equivalent plastic strains happened at the loading areas. Figure 5.31 depicts the load deflection curve of the problem which has been achieved in an incremental deformation controlled process. The curve is in agreement with the Wriggers et al. [1996] within a wide region. 218 "Hill Figure 5.29* Pinched cylinder: Deformed shape and outer-surface contour plot of equiva- lent plastic strains at w = 150 (left) and at w = 200 (right). plastc Btra • 0.6102 0.4882 0.3661 0.2441 0.1220 0.0045 0.0000 0.9978 0.7675 0.5373 0.3070 0.0768 0.0050 0.0036 Figure 5.30. Pinched cylinder: Deformed shape and outer-surface contour plot of equiva- lent plastic strains at w = 250 (left) and at W = 280 (right). 5A2.6. Free-flying multilayer plate with ply drop-offs This example establishes the capability and performance of the present solid-shell in modeling elastoplastic multilayer plates/shells with ply drop-offs. The same model for linear elastic material was demonstrated in Subsection 4.4.4. The geometry, the loading configuration and the time history loading amplitude are described in Figure 4.9. The mesh of a three-layer plate with ply drop-offs is shown in Figure 4.21 (i.e., L = 0.3m, W = 0.06m, and {t) h - 0.001m). The plate is divided into three equal parts and two ply drop- offs along its length, with each part having a length of 0. 1 . We use sixty solid-shell elements to model the three-layer plate with ply drop-offs, with a time-step size of At = 25 x 10~ 6 sec and a total simulation time of t = 0.08sec. The effect of gravity force is not considered. The material properties are E = 206. x 10 9 Pa, u = Q.? p= 7S00.Kg/m 3 , r y = 245 x 10 6 Pa, H - 1. X 10 9 Pa , (5.100) 3500 3000 2500 2000 I*. 1500 1000 500 50 100 150 200 250 300 W Figure 5.31. Pinched cylinder: Load deflection curve. where E, fi, p, r y , H are the Young's modulus, the Poisson's ratio, the mass density, the initial yield stress, and the isotropic hardening parameter, respectively. Snap shots of the multilayer plate undergoing large overall motion and large defor- mation are taken in every 4 x 10 _3 sec, from the simulation using the implicit Newmark algorithm without numerical dissipation, and are displayed in Figure 5.32. Furthermore, the variation of linear and angular momenta along the time are shown in Figure 5.34. Both the linear momentum and the angular momentum are conserved. On the left of Figure 5.35, the kinetic energy, the strain energy (computed via (5.93)), the total energy (kinetic + strain energies), and the work of the external forces are plotted as a function of the integration time. Due to the physical plastic dissipation, the total energy is much less than the work of the external forces, without any numerical damping. Unlike in the elastodynamics, the strain energy remains on a very low level, which indicates that the free-flying of plate is essentially a rigid-body movement (comparing Figure 5.33 to Figure 5.32). For the elasto- plastic dynamic analysis, the Newmark algorithm possesses a stable number of iterations ■ 220 and thus a good rate of convergence at each time step (right of Figure 5.35). The smooth- ness of both kinetic energy and strain energy along the time suggests a very low level of high-frequency "noises" in the response, compared to the elastic case where the introduc- tion of numerical dissipation is crucial to suppress the high-frequency modes. 1 "I- '' Figure 5.32. Free-flying three-layer plate with ply drop-offs using the Newmark algorithm: Perspective view. Figure 5.33. Free-flying three-layer plate with ply drop-offs: Deformed shapes for linear material material (left) and elastoplastic material (right) at t = 16 x 10 _3 sec. 221 Figure 5.34. Free-flying three-layer plate with ply drop-off s: Linear momentum and angu- lar momentum using the Newmark algorithm. Strain • KrnahoStrain - Ealamal Wor* 0.01 0.02 0.03 0.04 0.05 O.OB 0.07 0.06 Tmm(mc) Figure 5.35. Free-flying three-layer plate with ply drop-offs: Energy balance (left) and Number of iterations (right) using the Newmark algorithm. Overall, the Newmark method yields a stable integration for the elastoplastic shells. It is interesting to note that a more expensive mid-point rule may conserve the angular momentum exactly via a product formula algorithm, while the accuracy of the dynamic response remains about the same as in using the Newmark method (Simo [1992, Fig. 14]). 5.4.2.7. The impact of a boxbeam In this example, we simulate a boxbeam being impacted at one end while the other end being fixed. The rigid impactor is assumed having an infinite mass with a fixed velocity of v = 20m/ s (Figure 5.36). For this high-speed problem with rough response, the explicit method is suitable. 222 The material properties are £ = 21. X 10 9 Pa, n = 0.3, p = 7800. Kg/m 3 , r y = 1.06 x 10 9 Pa, H = 40.9 x 10 6 Pa , (5.101) where E, p, p, r y , H are the Young's modulus, the Poisson's ratio, the mass density, the initial yield stress, and the isotropic hardening parameter, respectively. Neglecting frictions between the impactor and the boxbeam, and without considering the self-contact on the surface of the boxbeam, we assumed that once the impact occurs, the rigid impactor stays with the impacted end of the boxbeam, which moves down with the same constant velocity v as the impactor, while its displacements in both X-direction, and V-direction are constrained. Due to the symmetry, only one quarter of the boxbeam is modeled by 960 solid-shell elements (2,080 nodes), as compared to 7,952 particles used in meshless method of Li et al. [2000]. With time-step size At = 0.4 x 10" 6 sec, a sequence of deformations are displayed in Figure 5.37-5.38, where half of the structure is displayed for a better visualization of the buckling modes. The experiment results show that the first few buckling modes should appear immediately at the impact location (Zeng and Combescure [1998]). Our results give the same prediction on the locations where the buckling modes should appear. The final deformed shape and the contour of equivalent plastic strains of outer surface of the boxbeam at t = 1.6 x 10~ 3 sec is shown in left of Figure 5.39. It is observed that the large equivalent plastic strains take place at the corner of the boxbeam. The relation between the reaction forces and the deflection at the collision end is shown in right of Figure 5.39. For the comparison, the calculation by using the implicit Newmark method was also carried out with time-step size At = 4 x 10" 6 sec, and similar results on deformations confirm the validity of explicit method with the present solid-shell element in elastoplastic large deformation shell analysis. 223 Rigid Impactor v = 20m/s L •■ 203mm a = 50.8mm b = 38.1mm h = 0.914mm Y X h Cross section Figure 5.36. The impact of a boxbeam: geometry and loading. Figure 5.37. The impact of a boxbeam: The deformation at time t = 0.4 x 10~ 3 sec (left) and at time t = 1.0 x 10" 3 sec (right). 5.4.2.8. Pipe whip This example is to show the modeling capability of the present solid-shell element in contact problems. The element provides a natural and efficient way for shell contact problem since double-side surfaces of shell are available and the transverse normal stress is included. To authors' knowledge, the shell contact problem using solid-shell elements Figure 5.38. The impact of a boxbeam: The deformation at time t = 1.3 X 10 _3 sec (left) and at time t = 1.6 x 10" 3 sec (right). piastic stra 0.4834 0.3443 0.2052 0.0662 0.0087 0.0005 Figure 5.39. The impact of a boxbeam: Deformed shape and outer-surface contour plot of equivalent plastic strains at time t = 1.6 X KT 3 sec (left), and deflection versus reaction force curve (right). has not been discussed in the literature before. This transient dynamic analysis simulates the impact of two steel pipes without con- sideration of friction. The pipes have the outside diameter of 3.3 1 25, the thickness of 0.432, and the length of 50. The target pipe is supported with a fixed boundary at each end. The impacting pipe swings freely about a point at one end with an angular velocity of 75 radians per second (Figure 5.40). The material properties for both pipes are the same E = 3.0 x 10 7 , p = 0.3 ,p = 7.324 x 10~ 4 , r y = 7.0 x 10 4 , H = , (5.102) where E, fx, p, Ty , H are the Young's modulus, the Poisson's ratio, the mass density, the initial yield stress, and the isotropic hardening parameter, respectively. Because of the symmetry, only one half of the geometry was modeled, in which total 840 solid-shell elements are used with one element in the thickness direction (1,830 nodes). Total 300 time steps are used with an equal time-step size At = 5.0 x 10" 5 sec in the implicit Newmark time integration without numerical dissipation. Slide surfaces are defined in potential contact outer surfaces of both pipes. This prevents nodes on one pipe from penetrating element surfaces of the other pipe, and allows the contact areas to evolve as the pipes deform. Here the slave surface is defined over the region of refined mesh at the outer surface of the impacting pipe, and master surface is defined over the region of refined mesh at the outer surface of the impacted pipe. Figures 5.41-5.42 shows deformed configurations at various time stages of the calcu- lation. The contact begins at a single point, and then evolves to two separate regions. Late in the analysis, a gap opens at the initial contact point. At the final state of t = 15 x 1(T 3 sec, the upper pipe has obviously rebounded. Figure 5.43 shows the contour of equivalent plas- tic strain at outer surface of both pipes at the final state of deformation, in which the cross section of the upper pipe is severely collapsed along with a local yielded region around the contact area. Initial angular vel. Figure 5.40. Pipe whip: Geometry and loading. Since this problem involves dynamics and contact between deformable bodies with 226 Figure 5.41 . Pipe whip: Deformed shape at t = 10 3 sec (left) and t = 5 x 10~ 3 sec (right). Figure 5.42. Pipe whip: Deformed shape at t = 10 x 10" 3 sec (left) and t = 15 x 10 _3 sec (right). nonlinear material, it provides a very good illustration of the applicability of the proposed solid-shell element to a wide range of problems. Figure 5.43. Pipe whip: Deformed shape and outer-surface contour plot of equivalent plastic strains, t = 15 x 10 -3 sec. CHAPTER 6 SOLID SHELL ELEMENT FOR ACTIVE PIEZOELECTRIC SHELL STRUCTURES AND ITS APPLICATIONS 6.1. Introduction An active shell structure has distributed sensors and actuators, along with the control algorithm to control the response of the host structure. The applications range from the vibration and buckling control (Berger, Gabbert, Koppe and Seeger [2000], Balamurugan and Narayanan [2001a]), to the shape control (Gabbert, Koppe and Seeger [2001]) and the noise suppression (Lim, Gopinathan, Varadan and Varadan [1999]). The active struc- tures have great potential in the design of light-weight and high-strength structures that are widely used in areas such as aerospace (Thirupathi, Seshu and Naganathan [1997], Loewy [1997]) and automotive industries (Chopra [1996]). In recent years, considerable effort has been devoted to the modeling and control- ling of active piezoelectric shell structure, see, e.g., Chee, Tong and Steven [1998], Sunar and Rao [1999], Benjeddou [2000] and references therein. The coupled electromechani- cal properties of piezoelectric ceramics and their availability in the thin shell form make them increasingly popular for the use as distributed sensors and actuators (Niezrecki, Brei, Balakrishnan and Moskalik [2001]). The direct and converse piezoelectric effects govern the electromechanical interaction in these materials. The direct piezoelectric effect states that a strain applied to the material is converted to an electric charge, while the converse piezoelectric effect states that an electric potential applied to the material is converted to a strain. The finite element modeling based on the classical laminate plate theory (Hwang and Park [1993]) and first-order shear deformation theories (e.g., Detwiler, Shen and Venkayya [1995], Balamurugan and Narayanan [2001/?]) has certain limitations due to their improper 228 229 modeling of the piezoelectric shell structure (Gopinathan, Varadan and Varadan [2000]). Improvements are made by using high-order shear deformation theories (Correia, Gomes, Suleman, Soares and Soares [2000]) and layerwise shell element formulations (Tzou and Ye [1996], Saravanos [1997]), while some shortcomings still remain in that 1) they do not consider the transverse normal stress in the element formulation, which may affect the be- havior of multilayer structure (Fox [2000]); 2) for the large deformation analysis, the finite rotation update associated with rotational degrees of freedom (dofs) in shell formulations is very complex to handle (Vu-Quoc, Deng and Tan [2000]); and 3) it is troublesome, if not impossible, to incorporate nonlinear piezoelectric material models associated with large input signals (e.g., Ghandi and Hagood [1997], Kamlah and Tsakmakis [1999], Kamlah and Bohle [2001]) into shell elements based on the plane-stress assumption. On the other hand, due to i) the complex geometry, ii) the material anisotropy, iii) the coupling of electric field and mechanical field, and iv) the need to satisfy boundary conditions of both electric field and mechanical field, 3-D detailed modeling for piezoelec- tric shell structures is used extensively. For example, twenty-node solid element in Koko, Smith and Orisamolu [1999] and eight-node solid element with incompatible modes in Ha, Keilers and Chang [1992] and Panahandeh, Cui and Kasper [1999] were used to model such structures. As mentioned in Chapter 3 and demonstrated here, the eight-node solid element with incompatible modes and the twenty-node solid element (with reduced integration) are not as accurate as shell elements for thin plate/shell problems. To achieve better accuracy, excessive number of solid elements are needed (see, e.g., Figure 2.24). To be computation- ally economical, Kim et al. [1997] has proposed the use of 3-D solid elements in modeling the piezoelectric devices, shell elements for the host structure, and transition elements to connect 3-D solid elements in the piezoelectric region with shell elements used for the structure. The modeling is complex to handle because of the mixing of several types of el- ements and the need for tuning of the aspect ratio of the transition elements. Moreover, the use of 3-D solid elements still lead to unnatural stiffening of piezoelectric devices, and as a result, artificially high natural frequencies. Alternatively, an assumed-stress piezoelectric 230 solid-shell element is proposed in Sze, Yao and Yi [2000], which had coarse mesh accuracy and could give accurate results for linear piezoelectric shell analysis; By construction, how- ever, the assumed stress method, obtained from the Hellinger-Reissner variational principle (Zienkiewicz and Taylor [2000a, p.284]), encounters the difficulty when incorporating the classical strain-driven nonlinear material models (Simo et al. [1989]). Therefore, the objective of our investigation is to develop efficient and accurate fi- nite elements which have the ability to model the nonlinear coupled mechanical-electrical response of multilayer composite shell structure which contains distributed piezoceramics. In the present work, a piezoelectric solid-shell element is formulated, and active vibration control of multilayer plate/shell with distributed piezoelectric sensors and actuators is re- alized by the control algorithm of the linear-quadratic regulator (LQR). The simplicity and efficiency of solid-shell elements for general nonlinear shell applications were proved in our previous work (Chapters 3-5). By including the electric dofs in the solid-shell ele- ment, the piezoelectric solid-shell element is developed within the context of the Fraeijs de Veubeke-Hu-Washizu (FHW) variational principle. Furthermore, a composite solid- shell element is proposed based on the solid-shell element previously developed, which can reduce considerably the number of equations, therefore, the amount of memory and the running-time, while at the same time keeping the same accuracy for thin multilayer composite shells and having the flexibility for the refinement through the thickness. The combination of the present piezoelectric solid-shell element and the solid-shell element en- ables the analysis of general curved active shell structures in the nonlinear regime, which in practice, becomes increasingly important such as in aerospace, MEMS applications. By combining the control algorithm, the sensor outputs predicted by the FE analysis could be used to determine the amount of input to the actuators for controlling the response of the integrated structures in a closed loop. The outline of this chapter is as follows: Section 6.2 reviews the kinematics and presents variational formulation of the piezoelectric solid-shell element, and discusses the composite solid-shell element. Section 6.3 presents the control design for structures 231 with piezoelectric sensors and actuators. Numerical simulations which illustrate the per- formance of the proposed formulations, including comparisons with available experiment results and solutions obtained from shell elements and solid elements, are given in Sec- tion 6.4. 6.2. The Solid-Shell Formulation We have developed the solid-shell formulation for geometric and material nonlinear- ities based on the enhancement of the Green-Lagrange strain E (see Chapter 3, Chapter 5), which leads to particularly efficient computational effort. We briefly describe below the kinematics of a piezoelectric solid shell in curvilinear coordinates, and present the piezo- electric solid-shell element formulation based on the three field FHW variational principle and the EAS method, and then discuss the composite solid-shell element. Readers are referred to Chapter 3 for more details. 6.2.1. The Kinematics of Piezoelectric Solid-Shell Formulation To overcome the known problems associated with the rotational degrees of freedom in the shell elements, we describe the shell kinematics by linear combination of a pair of material points at the top and bottom surfaces of the shell. Therefore, the assumption of the shell element that the normals to the element mid-surface remain straight but not necessarily normal during deformation, is still adopted. Thus, the initial (undeformed) three-dimensional continuum of the shell geometry (Figure 3.2) t = e □ := [-1,1] x [-1,1] x [-1,1] , (6.1) where X u and Xi are the position vectors on the upper surface and lower surface of solid- shell element in initial configuration, respectively, (£\ £ 2 ) the convective coordinates in the in-plane direction, £ 3 the convective coordinate in the thickness direction, and □ represents the bi-unit cube. Similarly, in the deformed configuration, the current three-dimensional continuum is 232 described by (6.2) where x u and x x are the current position vectors on the upper surface and the lower surface of the solid-shell element in the deformed configuration, respectively. The initial configuration is related to the deformed configuration (see Figure 3.3) by the displacement field u as x(0 = x(o + w(0 (6.3) The convected basis vectors in the initial configuration are obtained by partial deriva- tives of the position vector X with respected to the convective coordinates £' as dx(t) 1 = 1,2,3, (6.4) which satisfies G { • G j = 8{ , dj — Gi'Gj , i,j = l,2,3, (6.5) where Gij are the metric coefficients of the initial configuration. To simplify the presen- tation, we will omit (£) in vectors like Gi($). The covectors G 1 can be obtained by the following G l = G lJ Gj, with G 13 = [Gij] € i-l ,_ 10)3x3 (6.6) and the convected basis vectors in the deformed configuration are defined in the similar way, by using (6.3) and (6.4), dx du g 1 = -^- i =G i + — , 2 = 1,2,3, (6.7) similar to (6.5) and (6.6), the covector g l , the metric coefficients g%j, and g lj in the deformed configuration are obtained. 233 By using (6.4) and (6.7), the deformation gradient F relates the basis vectors G t of the initial configuration to the basis vectors g { of the deformed configuration as follows F=^r=^®GT, (6.8) and with (6.5) and (6.8), the compatible Green-Lagrange strain tensor E c is obtained as E c = l - (F T F ~h) = \ (9ij ~ Gij) G 1 & - E\p G j , (6.9) where J 2 is a second-order metric tensor, Ef s are the covariant components of the strain tensor E c . The corresponding second Piola-Kirchhoff stress tensor S is expressed in the same convected basis G iy i.e. S = S %j G l ®G j . (6.10) where 5 lJ are the covariant components of the stress tensor S. Similar to (6.1), the electric potential <j> is described as follows m = \[(i+e)<i>u{e,e)+{i-e)^Ke)] , where </> u and fa are the electric potentials on the upper surface and lower surface of solid- shell element, respectively. The electric field £ is derived from the gradient of the electric potential (p with respect to the position vector X as 5(0) = -GRAD(0) = -^G l . (6.12) The corresponding electric displacement vector D is expressed in the same convected basis Gi D = D l Gi . (6.13) 234 6.2.2. Piezoelectric Solid-Shell Element Here we present the piezoelectric solid-shell element formulation used for piezoelec- tric sensors and actuators in active shell structures. The variational formulation of the EAS method, and the finite element approximation of the developed solid-shell elements for the multilayer composite shells are described in Chapter 3. 6.2.2. 1. Functional and finite element formulation The electric flux conservation for the piezoelectrics is described as follows Yl E = _ J D-SdV - J q v 4>dV - J qsHS , (6-14) Bo Bo S q where D is the electric displacement vector, S the electric field defined in (6.12), 4> the electric potential, q s the electric flux per unit area on the surface, and q v the electric flux per unit volume. The variation of functional (6.14) with respect to the electric potential is SU E = - J D'SEdV- J qvHdV- J q s 5(pdS, (6.15) Bo B S q and the variation of the functional of mechanical energy U M was presented in Section 3.3 and rewritten as follows : J S : 8EdV - j 5wb*pdV - J 5wt*dS . (6.16) Bo Bo S a where the enhanced Green-Lagrange strain E = E c (u) + E (a). The total variation is the combination of (6.15) and (6.16) 8U^ = fllg + 6n { * ] = 8U sttff + 5Yl ext , (6.17) in which 5Il sm = J S : 5EdV - J D'SEdV , (6.18) Bo Bo SKext = ~ J 6u-b*pdV - J 5wt*dS - 1 8<t>q v dV - j 6(f>q s dS , (6.19) B S a B 235 where the second Piola-Kirchhoff stress S and the electric displacement D are determined by the constitutive law with the knowns E and £. Following the standard finite element discretization as explained in Hughes [1987] or Bathe [1996], we discretize the reference configuration B with numnp nodes and nel elements, i.e., the discretization of the reference configuration B into a collection of finite I 1 ne ' (e) element subdomains Bn , such that Bo ~ U B . e=l Besides the same interpolations used on displacements u and Green -Lagrange strains E°, E in Section 3.3, additional linear interpolation is used on the electric potential 0, and the variation and increment of electric field £ by denoting * iVV e) , {Mi} = -B*Sf {e) , {AS Z } = -B*A0 (e) , (6.20) where AT* and B 4 ' take similar forms as in the Appendix A.l. We apply a standard finite element procedure to the discrete weak form (6. 17) on the element domain £?q 6ii^ = sn$ + su% ] = m% + = o , (6.2 n where the discrete weak forms of the stiffness operator 511 and the external forces 511^ become * n Kf = / * {Etf {S lj } dV + J 6 {E^Y {S^} dV - J 6{S 1 } T {^}dV, t$> (6.22) *n2 = - J Su-b*pdV- J 6wt*dS- J S(j)q v dV - J Scf) q s dS , (6.23) B (e) 5 W B («) s (e) where {S lj } is the vector of the second Piola-Kirchhoff stresses in the order of {S ij } = [S u , S 22 , S 12 , S 33 , S 23 , S U } T , (6.24) and the vector of electric field {£J is as follows {Si} = [S U S 2 ,S 3 f , (6.25) 236 and the vector of electric displacement {D 1 } is given as {&} = [D\D 2 ,D 3 ] T . (6.26) To simplify the presentation, we will omit the iteration index k, in the following lineariza- tion procedure. The linearization of (6.21) with respect to the primary unknowns (d^ e \ <pt e \ a.^) is V (AIM) • (AS e \ A^\ Aa«) = - (SU% + 611%) , (6.27) where the variations SU.^ in (6.22) and 5U% in (6.23) are m% («*•>, ^\a^) = 5d^ T f% stlJ} + 5^ T ff stlS + fa^/M, , (6.28) with/S^= / B T {S*}dV, f% tiff = J B* T {D>}dV, (6.29) fEAs= J Q T {S tj }dV, (6.30) (d« 0< e >) = -UPrftyt - 6f>MU , (6.31) with/JL = / N T b*pdV+ J N T t*dS, (6.32) /SL- / N* T q v pdV + J N* T q s dS, (6.33) «( e ) <?(«) by linearizing (6.28) and (6.31), the left hand side of (6.27) becomes V (<m (e) ) • (AS e \ A^ e \ AqW) = 5d^ T (fcWAdW + fc#A^ + fc^AaW) +*0« r (fcg Ad« + fcW a^J + fcgAaW) +5a< e > r Ad« + fcg A0 (e) + fcgAaW) . (6.34) 237 Defining the matrices of tangent material moduli C, e, e T , e as C = Qij Id e = Mi e = dE kl dD k dE~j 'dS ij d€ k dD i d6x6 d 3x6 d6x3 d3x3 (6.35) (6.36) (6.37) (6.38) where material moduli C, e, e T , e are expressed in the convected basis and subsequently arranged in matrices according to the ordering of the stress components in (6.24) and the electric field components in (6.25). From (6.29)!, (6.35) and (6.37), it follows that °J Mstiff = = I ( G " s + BT ° B ) dV - »(e) M - df (e) Mstiff d<f> = J B T e T B*dV , «<«) *~ - S - / BT ° gdv • b(«) (6.39) (6.40) (6.41) where the matrix G, the stress matrix S>, and the enhancing strain interpolation matrix Q are given in Section 3.4. From (6.30), (6.35), and (6.37), it follows that fcto (e) 5/ uot =1* T CBdV B ■ (e) df dcf) W .(«) = / g T e T B 4 'dV , (e) £45 5/ (6.42) (6.43) (6.44) 238 From (6.29)2, (6.36), and (6.38), it follows that K 4>u — df to Estiff dd fc to / B^eBdV , ,(e) fc to 0e* pto .(e) _ ^ firtijf ®f ^Estiff dato k to ■/ «<«) B^eQdV , >4> 90 to «(«> B* T eB*dV . (6.45) (6.46) (6.47) By substituting (6.30), (6.32), (6.33) and (6.34) into (6.27), the linearized system of equations on the element level is obtained for arbitrary variations of the element nodal dis- placement d (e) , the element nodal potential d^ e \ and the element local internal parameters aS^ as follows fc«Ad« + k^A^ + ftgAaW = , W ;to (6.48) (6.49) (6.50) Vu^- 1 "- *~ t "-^q^- 1 " — y e i( J Estiff ■> ^0 Since the enhancing strains J5 are chosen discontinuously across the element bound- aries, the elimination of the local internal parameters at the element level is possible. From (6.49), the increment of internal variable vector can be expressed as Aa« = - \ki e l (6.51) then substituting (6.51) to (6.48) and (6.50), the condensed system of equations in element (e) is as follows Ad^ A0 (e) where the condensed tangent stiffness matrices -(e) -(e) K uu K u<t> -(e) -(e) K <j>u K <P<p J .w M .to iu (e) _ ( k (e) _ k (e) \ k (e)]- l k (e) ^<t>4> — \ K <H> K 4>a [ K aa\ K a4> ) i k ie) = au I > n ud> k k (e) - fc (e) \k (6.52) .to "a<p j 1 (6.53) 239 and the condensed element residual force vectors r M — J ext J Mstiff T ^ua k r (e) - n - f (e) 4- ' £ — Hext J Estiff ' ^(pa k e) -1 EAS > t(e) EAS (6.54) After assembling the element matrices in (6.53) and element residual force vectors in (6.54), we obtain the incremental displacement-potential problem as follows K uu K u<j> Ad Rm Re (6.55) (6.56) (6.57) with K uu = Ak ( 2 , = [K^f = KkZ , K H =Ak e=l e=l T e=i nei nei Rm = Jkr M ] , = Arg* , e=l e=l where the action of the assembly algorithm is denoted by the assembly operator A. For the nonlinear dynamic response calculation, the implicit time integration schemes can be employed. The incremental dynamic equations are obtained by including the weak form of inertial forces and its linearization in (6.21) and (6.27) respectively, we refer readers to Section 4.2 for the details. 6.2.2.2. Linear piezoelectric material law in convected coordinate The constitutive relation for linear piezoelectrics is expressed as S = C *. E-e T 'S , D = e : E + e'£ , (6.58) (6.59) where the stress tensor S are expressed in the Cartesian basis a* and convected basis G { as S = S ab a a ® a b = S l3 G 1 <8> Gj , we obtained the relation between components and S ab (6.60) (6.61) 240 where G x -Gj - Sjj, o, = a 1 . The strain tensor E is expressed in different basis a, and Gj as E = E cd a c ®a d = E kl G k ® G ; , (6.62) we obtained the relation between the components E cd and E kl E cd =(G k -a c ) {G l -a d )E kl . (6.63) The electric displacement vector D is expressed as D = D l Gi = D a a a , (6.64) thus the relation between the components D l and D a is D x = (G l -a a )b a . (6.65) The electric potential gradient vector £ is expressed as S = = £ a a a , (6.66) thus the relation between the components t\ and 8 a is £ a = (a a >G l )£ i . (6.67) The fourth-order elastic constitutive tensor C is as follows C = C abcd a a ®a b ®a c ®a d = C ijkl G t ® G^ ®G k ®G[, (6.68) and the component form is The third-order piezoelectric tensor e determined at constant strain takes the form e = e abc a a ® a 6 ® a c = e y *G< <g) G, ® G fc , (6.70) 241 and the component form is the following e ijk = (G l -a a ) (G j -a b ) (G k -a c ) e abc (6.71) and the second-order dielectric tensor e determined at constant strain is expressed as e = e ab a a ®a b = 6 l3 Gi®Gj , (6.72) thus the component form is e««(G i -o.) (&-a b )e ab (6.73) From (6.58) and (6.59), the relationship between the stress components and the strain com- ponents, the electric displacement components and the electric field components, with re- spect to different basis a* and G { are, respectively J = O Hied — e &m ) iabcd mab i D a = e abc E bc + e ab £ b , ~ab i (6.74) (6.75) and gn = c ljkl E kl -e m] £ n , D l = e ijk E jk + e% . (6.76) (6.77) If we express (6.61), (6.63), (6.65), and (6.67) in the matrix form by the same com- ponent ordering as in (6.24) and (6.25), we obtain {&} - T T G {S ab } , {E cd } = T G {E kl } , {D 1 } = T T e {/>} , {£ a } = T e {£} , where the matrix T G is in Section 3.3, and the matrix T e is T = t 1 t 2 t 3 L \ L l L \ A f 2 f 3 b 2 l 2 L 2 L ^3 ^3 ^3 . (6.78) (6.79) (6.80) 242 with the coefficients t{ = a, , G- J , i, j = 1, 2, 3. Since in the Cartesian coordinate, the constitutive relation of (6.74) and (6.75) in the vector form are expressed as follows {S ab } = [c abcd ] {E cd } - [e mab ] {S m } , {/>} = [e abc ] {E bc } + [e ab ] {£ b } , (6.81) (6.82) and in the convected coordinate, the constitutive relation of (6.76) and (6.77) in the vector form is as follows {E jk } + e« {Ej} , (6.83) (6.84) substituting (6.78) 2 and (6.79) 2 into (6.81) and (6.82), and then substituting (6.81) and (6.82) in (6.78)i and (6.79) r . By comparing the resulting equations with (6.83) and (6.84), the constitutive matrix Qijkl , and [e lj ] in the convective coordinates associated with basis G { are transformed from the ones in the Cartesian coordinate ,ijk = T = T = T C e abc -ab j, abed G (6.85) (6.86) (6.87) 6.2.3. Composite Solid-Shell Element To improve the modeling efficiency of laminated composite shells, a composite solid- shell element is developed here. As shown in Figure 6.1, the material layer can be stacked in parallel to the upper surface and lower surface of the eight-node solid-shell element. The element matrices are obtained by using the numerical integration, in which 2x2 Gauss quadrature is used in the plane of the lamina, and two Gaussian points are used for each material layer in the thickness direction. H c3 1 s C / G / B ///// A //// / w- ■ e x layer n x layer i layer 1 243 Figure 6. 1 . Composite solid-shell element: Eight-node composite solid-shell element in the isoparametric space, two Gaussian points (x) for each layer, and eight collocation points for assumed natural strain methods. Assuming that the thickness of layer (£) within an element remains constant n and total thickness H = of n layers is much smaller than other dimensions of the element, it is straightforward to find isoparametric coordinates £ 3 in the thickness direction of Gaussian points at each layer. For a typical layer (i), the isoparametric coordinates and in the thickness direction at its lower surface and upper surface are respectively i-l 2 » (6.88) 2 1-1 2 H: By using and the isoparametric coordinates of layer Gaussian points through the thickness direction are obtained. Accordingly, the weight factor for each integra- tion point at layer (i) is scaled by (i) h H To avoid shear-locking of the displacement formulation, we employ the assumed natural strain method, as applied to the four-node shell element in Dvorkin and Bathe [1984]. The assumed transverse shear strains are based on the the constant-linear interpo- lations of compatible transverse shear strains E^ 3 , E% 3 in (6.9), evaluated at the midpoints Q = A, B, C, D of the element boundaries with £ 3 = (Figure 6.1). In the case of curved thin shell structures or the nonlinear analysis, to circumvent the locking effect from parasitic transverse normal strain, we employ an assumed strain approximation for the covariant component E$ 3 of the compatible Green-Lagrangian strain 244 tensor, refer to Betsch and Stein [1995]. We assume bilinear interpolations of the transverse normal strain field, where the points Q = E,F,G,H at the corners of element midsurface (Figure 6.1) serve as sampling points of the compatible transverse normal strain. The readers are referred to Section 3.3 for more details. 6.3. Simulation Control Design In this section, we present the procedure in integrating the finite element analysis with the control algorithm to simulate and control the response of an active structure with piezoelectric sensors and actuators (Figure 6.2). The static condensation is employed to eliminate the zero-mass degrees of freedom (dofs) associated with the electric field. A modal analysis is then performed to transform the coupled finite element equations of mo- tion into the reduced-order model in the modal coordinates. The linear quadratic regulator (LQR) is then employed to emulate the optimal controller by solving the Riccati equations from the modal state space model. Feedback 4> a Actuator Sensor Host structure Controller Sensor 4> s Figure 6.2. The typical active structure configuration. 6.3.1. Finite Element System Equation of Piezoelectric Structure The finite element equation for the linear piezoelectric structure, without considering damping, is as follows M d + K<j>u d Fext Qext (6.89) where stiffness matrices K uu , K u4> and are in (6.56), and external forces F ext and Q ext are from (6.57) without considering the internal forces, M the mass matrix. Note that the dofs with the electric field are "massless." 245 We partition the piezoelectric dofs into two parts, that is, sensor dofs and actuator dofs, as follows = S 0° l Qext T s ext Q o a ^ext (6.90) where the actuator voltage dofs (p a are known from the input, and the external electric flux Qe Xt associated with the sensor dofs, in general, is zero; the voltage dofs (f> 3 and electric flux Q" xt are unknowns. The matrices and K U(t> become K<j>4> — <t>4> ryss jy-aa i — [K U( p} T — jy~ss jy-aa 4>u ** q>u where K S A. = r jyss ryaa u <t>. . and JC" = Accordingly, the system equations of (6.89) becomes MOO d 1 F ext ?• ► + TS-SS K u H . ii 0u 44 iv-aa K 4><t> J From (6.93)2, the sensor voltage increments <j> s are obtained by p = (Qext -K^d-K^) substituting (6.94) into (6.93) a , we obtain ryss M d + [K uu -K s ° ext *» Qext + -K"u! ryss -1 (6.91) (6.92) (6.93) (6.94) «5 - *C </> a • (6.95) During the solving procedure, we calculate the displacements d by (6.95), then solve the sensor nodal voltage 4> s by (6.94). 246 There are two particular cases in (6.93), that is, only sensors existing or only actuators existing in the structure. If only sensors exist (i.e, <f> = <j> s ), the system equation (6.93) is reduced to M " {IV n uu u<t> jy-ss IS" s s ■ fY 4>u ■** 4>4> d S Fext Qext (6.96) we can solve for d by setting = in (6.95), that is -l M d + K uu - Kl K S j?„ ) d = F ext — K. then solve for <p s by setting a = in (6.94), that is ' U(j> jy-SS 1 -1 Q ext ' (6.97) 1 (Qext -^d) (6.98) On the other hand, if there are only actuators in the structure (i.e, <f> = (f> a ), the system equation (6.93) becomes M d + T/~ ry-aa *» 1MI ■*» u<t> ry-aa -ry-aa d 4> a Fext Qext (6.99) we use (6.99)i to solve for d with the known <p a , that is M d + iir uu cZ = F ext - K™$ a , (6.100) then solve for the electric flux Q^ xt with (6.99) 2 , that is Q a ext = K a 4> a u d + K™<f> a . (6.101) 6.3.2. Reduced-Order Model of Piezoelectric Finite Element System Consider a typical structure bonded with piezoelectric actuators and sensors. The goal of the design is to suppress unwanted vibrations and increase damping of the structure, which can be achieved by proper controlling of voltage signals 4> a to the actuators. With the external electric flux Q s ext = 0, (6.95) and (6.94) become respectively M d + [ K uu - K lie? ry-ss K 4>4> K'L d = F ext + K u<p ry-ss 4><t> -l (-K£d - K™<j> a ) , (6.102) (6.103) 247 To compute the signal outputs s from piezoelectric sensors, we assume that there is no coupling between sensors and actuators such that in (6.102) and (6.103) K£«0, (6.104) and denote l K%) ■ (6.105) At the right hand side of (6.102), the actuation force vector F a , is related to the coupling stiffness matrix JK^J and the vector of applied voltages <p a by F a =-K%<p. (6.106) Using (6.104), (6.105), and (6.106) in (6.102) and (6.103), the governing dynamic equa- tions of the structure under both mechanical excitations and actuation forces can be ex- pressed as follows Md + Kd = F ext - F a , (6.107) and the sensor output is l K&d- (6.108) It is noted that the response of the system in (6.107) is regulated by the control voltage vector (f) a , which depends on the information of the states of the system measured through sensors. To construct a control law much more efficiently, all matrices in (6.107) should be transformed into diagonal forms. For the linear system, (6.107) can be decoupled by means of the modal transformation, which is based on the solution of the generalized eigenvalue problem X* = M*0 2 GR nxn , (6.109) where Vt 2 = diag [uf, •••,u$ € contains the eigenvalues, and # = [rjf v ■ • • , <0J e R nxn contains the eigenvectors. K=[K TfSS *u<t> K The cost of solving the generalized eigenvalue problem (6. 109), however, can be pro- hibitively high for the large size n. For structural dynamics problems, the typical modal analysis studies on the frequency content and spatial distribution of the excitation have shown that the response is controlled by a relatively small number of low frequency modes. On the other hand, the finite element analysis approximates the lowest frequencies and the associated mode shapes best, and has worse accuracy in higher frequencies and mode shapes. Therefore, in practice only the mode shapes with low frequencies are used for the dynamic response of the structure. Two iteration methods, WYD Ritz vector approach (Wilson, Yuan and Dickens [1982], Leger [1986]) and Lanczos approach (Leger [1986], Parlett [1998]), can generate a few eigenpairs in low frequencies much more effectively, compared to the traditional subspace iteration method (e.g., Hughes [1987, p.576]). Both algorithms were implemented by authors in the code of CFD-ACE+ and listed in the Ap- pendix B.l. For the details of the performance of the above two algorithms, please see the series reports by Vu-Quoc, Tan and Zhai [1998-2000]. The transformation is given by d = <&q » V r q r , d = « y r q r , (6.110) where * is the modal matrix and q the vector of modal coordinates for the full-order model (6.107), \I> r and q r contain the first n r (<C n) modes and the corresponding modal coordinates. The eigenvectors Vl/ r are orthogonal with respect to both the stiffness matrix K and the mass matrix M, and normalized to M as follows VjKV r = Q 2 r , *jM* r = / , (6.1 1 1) where 0, = diag u/jf, • • • ,uj 2 t contains the first n r eigenvalues cuf of the structure. By substituting (6.110) in (6.107), and premultiplying at both sides of (6.107), and then using (6.1 1 1), we obtained the uncoupled reduced-order model q T + Q 2 r q r = <i> T r F ext + f r , (6. 1 1 2) 249 where f T is the modal control force vector as follows f r =*l F a = -V 1 r K™<t> a (6.113) In general, it is not necessary to control all the first n r modes in the reduced-order uncoupled model of (6.112). Instead, only the first few modes are used for the control design. For this purpose, the n r modal equations in (6.1 12) are separated into the first n c equations of controlled modes and the rest n u (= n r — n c ) equations of uncontrolled modes. Accordingly, the modal matrix ty r , the modal coordinate vector q r , the modal control force vector f r , and the modal excitation force vector & r T F ext are partitioned into two parts as follows Qc therefore, the dynamic equation for the first n c controlled modes are ) r ext * r F » x ext (6.114) , (6.115) q c + n 2 c q c = f c +* T c F ext , (6.116) where f2 c 2 = diag n u modes are , and the dynamic equation for the rest of the uncontrolled q u +n 2 u q u = f u +*lF ext , (6.117) where f2„ = diag UJ. n c +l> 6.3.3. Controller Design The controller or control law describes the algorithm or signal processing used by the control process to generate the actuator signals from the sensor and command signals it receives. Since the 1960s, modern state-space controller design methods such as the linear quadratic regulator (LQR) and the linear quadratic Gaussian (LQG) (e.g., Anderson and Moore [1990]) have been developed in the linear time-invariant (LTI) system. In the 250 uncoupled system (6.1 16), the linear-quadratic regulator (LQR) is used to determine the modal control force for any given mode, which depends on only the modal coordinate and modal velocity of that mode. As a result, the independence of the open-loop equations for each mode is preserved for the closed-loop system. For the details, readers refer to e.g. Boyd and Barratt [1991]. Letting Qc he (6.118) 2n c xl one can transfer the dynamic equations (6. 1 1 6) for controlled modes into state space equa- tions that are suitable for control design. The state space equations from (6.1 16) can be written in the matrix form as X c = AX c + Bf c + D F ext , (6.119) where A = / ■nl o B I D = (6.120) For the system described by (6.1 19), the linear feedback control law is defined as follows f c — - Kc X c , (6.121) where matrix K c £ K n = x2ric j s partitioned into the displacement gain matrix G d € R ncXnc corresponding to modal coordinates q c and the velocity gain matrix G v 6 ]R n cX«c corresponding to modal velocities q c K c =[G d G v ] , (6.122) substituting (6.121) into (6.1 19), it yields the closed-loop state-space equation X c = (A - B K c ) X c + D F ext , (6.123) for the response of the active control system. In terms of modal coordinates of controlled modes, the control force vector f c in (6.1 15) i and (6.121) takes the form f c = -<H c T K a u a A a = -K c X c =-(G d q c + G v q c ) . (6.124) 251 The optimal regulator state feedback gain matrix K c may be obtained by minimizing a linear-quadratic cost function J, which is defined as follows J = J™ (X C T QX C + f c T R f c )dt, (6. 1 25) where f c is the input force in (6. 1 24), Q the weight for the effectiveness, and R the weight for forces f c . The weighting matrix Q in the performance function J of (6.125) is usually assumed to be diagonal (Meirovitch [1990]) as follows Q m diag (wj, • • -,u 2 nc , 1, • • • , l) € R^ eX 2n e f (6.126) however, there are no general guidelines for the choice of R, the weighting matrix for control forces f c . A diagonal R, as assumed in many applications, may be used as R = diag (R u ■ ■ ■ , RrJ € M" cXnc . (6.127) Define the optimal gain matrix K c K C =R 1 B T P C , (6.128) which is obtained once P c is solved from the following Riccati equation A T P C + P C A + Q-P c BR 1 B T P c = 0, (6.129) where the above matrices A and B are from (6.120). With matrix K c in hand, the closed-loop response of the system is obtained by integrating (6.123). Subsequently, the control voltages cj) a for the actuators are calculated through (6.124) 4> a = [V/KZ]- 1 K C X C = [tf/lCj -1 (G d q c + G v q c ) , (6.130) where the number of controlled modes n c and the size of vector cjf of actuators may not be the same, hence the inverse, ^^K^ 1 in (6.130) is usually operated by a pseudo- inverse process. Therefore, the modal forces f c are only approximately independent, depending on how close ty/K^ is to be a square matrix. 252 The response of the structure is calculated by combining of the contribution from the controlled modes * c and uncontrolled modes * u d a y c q c + <H u q u , (6.131) the sensor voltages 4> s are then approximated by using (6.108) and (6.131) s « - [K%] _1 K s ; u (* c g c + * u qJ , (6.132) and the rate of the sensor voltages <p s are obtained by differentiating (6.132) with respect to time >*-[«Sr 1 ^S(»e4. + »«M • (6.133) Under a given structural excitation, the structure would vibrate accordingly, and the distributed sensor outputs could be calculated from (6.132). Then the voltage supplied to actuators could be determined from the control law by using the calculated sensor outputs as input. Then the new state of the structure could be calculated under both the external excitations and the actuation voltages applied to the structure through actuators. The pa- rameters in (6. 127) of the controller is then modified to optimize (or tune) the performance for the desired closed-loop response. From the closed-loop system of (6.123), it is clear that the displacement feedback - G d q c modifies the stiffness, and the rate feedback of - G v q c modifies the damping of the open-loop system (6.120), that is A - B K c / Ql — Gh — G, (6.134) The solving procedure on the above control design is given in the Appendix B.2. 6.4. Numerical Examples The finite element formulations of the present low-order solid-shell element for anal- ysis of piezoelectric shell structure presented in previous sections have been implemented in the Matlab, and run on a Compaq Alpha workstation with UNIX OSF1 V5.0 910 oper- ating system. In each element, the mass matrix is evaluated by using the Gauss integration 253 2 x 2 x 2, the tangent stiffness matrix and the dynamic residual force vector are evaluated by using the Gauss integration 2 x 2 in the in-plane direction, and two Gauss points in the thickness direction for each material layer. Below we present the examples involving the static, dynamic analyses and active vibration control of piezoelectric shell structures. 6.4.1. Cantilever Plate: Out-of-Plane Bending We present this example to show that the eight-node solid element with incompatible modes and the high-order twenty-node solid element (with reduced integration) suffer from the shear-locking in thin shell applications. A cantilever plate of length L = 10 and width W = 1 is subjected to the transverse shear loading F at the free end (Figure 3.10 and Figure 3.12). We consider various models with different aspect ratios L/h to compare the performances of several elements. To have the same level of deflection magnitude regardless of the thickness h, the applied loading F is set to be proportional to the thickness raised to power three (i.e., h 3 ) in the numerical calculation. The material properties are prescribed to be £ = 1.0xl0 7 , i/ = 0.4, (6.135) where E and u are the Young's modulus and the Poisson's ratio, respectively. A comparison of the tip deflection of the plate for different elements is shown in Figure 6.3. All results are normalized to the solution obtained from the geometrically- exact shell element (Vu-Quoc, Deng and Tan [2000]). The FE models involved are made of ten eight-node current solid-shell elements, ten eight-node solid elements with full in- tegration, ten eight-node solid elements with incompatible modes (Taylor et al. [1976]), five twenty-node solid elements with full integration, and five twenty-node solid elements with 14-point reduced integration scheme (Hoit and Krishnamurthy [1995]), all with one element in the thickness. It is noted that the direct use of twenty-node solid element with 2x2x2 reduced integration scheme encounters the singularity in this problem. In the linear problem, the transverse loading F = 10 4 h 3 is applied at the free end. The free-tip 254 transverse displacement along the force direction at the corner of the midsurface of the plate obtained from the geometrically-exact shell element, approaches w = 3.9236 at the thin limit (the solution from Euler-Bernoulli beam theory w = 4). For the large deforma- tion case, the loading F = 5 x 10 4 /i 3 is applied in five equal load steps. The tip deflection in the transverse direction obtained from the geometrically-exact shell element approaches w — 7.41366 at the thin limit. In both linear and nonlinear situations, the present solid- shell element yields excellent results even for the extremely thin plate (aspect ratio = 6667 or h = 1.5 X 10" 3 ). The eight-node solid element with full integration suffers from the se- vere locking. The twenty-node element (with reduced integration) and the eight-node solid element with incompatible modes cannot obtain correct results, especially for the large de- formation analysis, while the twenty-node element with 14 integration points had better performance than the eight-node solid element with incompatible modes and the twenty- node element with full integration. It is interesting to note that a twenty-node solid element with another 14 integration points rule by Irons [1971] gives the same unsatisfactory results as that with the full Gaussian integration (3x3x3 integration points), but taking almost half of the computational effort. - 5 20- noils bnck w/ 14 Rl - 10 8- nod* bock with Incomparibte mod* - 5 20-nodabnckw'FI - IQB-nod. brtdcw/FI 9 0.6 - -O -e-o 10 geometTKatfy sua si -<> 10 pressol element 520-node brkk w/ >4RI lOB-nodePrx 5 20-nod* bnck w/FI -*- iOfl-noo»bne> Lib. " '° * * L/h Figure 6.3. Out-of-plane bending: Element performance at different aspect ratios L/h for both linear case (left) and geometrical nonlinear case (right). For flat plates undergoing small deformation, numerical results show that it is suffi- cient to consider the ANS treatment of only the transverse shear strain E 13 and E 23 ; the additional ANS treatment of the transverse normal strain E 33 does not change the numerical results. 255 6.4.2. Multilayer Composite Hyperbolical Shell This examples is used here to verify the correctness of the proposed composite solid- shell element in thin shell applications. The same example was considered in Basar et al. [1993] to test their shell element formulation. The shell structure consists of three layers with the same layer thickness y)h = h/3, which are placed symmetrically with respect to the middle surface. Due to symmetry, only one-eighth of the shell structure is modeled with FE meshes of 14 x 14 x 3 solid-shell elements (one element per material layer) and 14 x 14 x 1 (one element through the thickness h) composite solid-shell elements respectively (Figure 6.4, 0° along circumferential direction). The layer material properties are E n = 40 x 10 9 , E 22 = £ 33 = 10 9 , u l2 = 1/13 = ^23 = 0.25 , G l2 = G u = G 23 = 0.6 x 10 9 . The analysis was carried out for two different stacking sequences: [0°/90°/0°] and [90°/0°/90°]. The load-displacement diagram Figure 6.5 shows that results obtained from the model with composite solid-shell elements (1,260 equations totally) agree with the refined model by having one solid-shell element for each material layer (2,520 equations totally). Therefore, the composite solid-shell element is accurate and more efficient to capture the overall global response such as the deflection for thin shells. The shell with the [90°/0°/90°] stacking sequence has larger deformation, and is less resistant to the loading than the shell with the [0°/90°/0°] stacking sequence. The computed results agree with those in Basar et al. [1993], where a layerwise shell element with complex rotation update was employed. It is noted that for the shell with the [90°/0°/90°] stacking sequence, a more refined mesh is needed to achieve the converged results (see also Basar et al. [1993]). The deformed shapes in Figure 6.6 for the final load P = 160 x 10 3 demonstrate clearly that large rotations and displacements are involved in this example. It is noted that for moderately thick shells or with nonlinear materials, the model with one element through thickness is incapable of accurately determining the structure response such as in-plane displacements, transverse shear stresses (e.g., Vu-Quoc, Tan and Mok [2002]). 256 Figure 6.4. Pinched multilayer composite hyperbolical shell: Undeformed mesh. 160 120 100 60 40 20 — ' f- T - - v(B) 0/90/0 9 + i - u (A) 0/90/0 -Q v(B) 90/0/90 + * u(A) 90/0/90 -* - v(B) 0/90/0 composite * u(A) 0/90/0 composite 6 i + * □ v(B) 90/0/90 composite 9 1 + n + • + u(A) 90/0/90 composite 6 i i <p d + k 6 J \ \ i 9 * + <t 6 I i 1 + « 9 f 6 i + 1 1 9 f + 6 I k \ i / 9 f 6 i 1 T + • <Z> f t J / ++ ++ V » i i Displacement Figure 6.5. Pinched multilayer composite hyperbolical shell: Load-displacement diagrams from both solid-shell elements and composite solid-shell elements, v(B) is the displace- ment along axis Y at point B, u(A) the displacement along axis X at point A. 6.4.3. Piezoelectric Bimorph Beam This numerical application is used to validate the developed piezoelectric solid-shell element in both an actuating and a sensing mechanism. The experiment consists of a can- 257 Figure 6.6. Pinched multilayer composite hyperbolical shell: Deformed shape with stack- ing sequence [0 o /90 o /0°] (left) and [90°/0°/90°] (right). tilevered piezoelectric bimorph beam with two equal polyvinylidene fluoride (PVDF) lay- ers bonded together, and polarized in parallel or anti-parallel directions, with the dimen- sions indicated in Figure 6.7. The beam is discretized into 10 equal solid-shell elements. The mechanical and piezoelectric properties of the PVDF are E = 2. x 10 9 Pa , v = 0. , e 3 i = e 32 = 0.0460C/m 2 , (6.136) Pn = V22 = P33 = 0.1062 x 10~ 9 F/m , where E and v are the Young's modulus and the Poisson's ratio, e 31 and e 32 are the piezo- electric stress coefficients, and p u ,p 2 2, and p 33 the electric permittivity coefficients, clamped Figure 6.7. Piezoelectric bimorph beam: geometry and mesh (left), and electric loading for anti-parallel polarization type (a) and parallel polarization type (b) (right). 258 For the anti-parallel polarization type (Figure 6.7(a)) and parallel polarization type (Fig- ure 6.7(b)), the theoretical results on the transverse deflection of the free-tip (Andersson and Sjogren [2001]) are calculated by 3 e 3 i V vl Wa= 2EV X w b = 3 (6.137) respectively, where h is the beam thickness, V the voltage applied on the surfaces of the PVDF layers. The present results obtained from both cases agree exactly with the theoretical re- sults. The deflections along the beam length for the anti-parallel polarization case are given in Table 6.1, in which the results obtained from the present piezoelectric solid-shell element are compared with that of a four-node shell element (Detwiler et al. [1995]), a nine-node shell element (Balamurugan and Narayanan [200 1&]), and the experiment (Ha et al. [1992]). Table 6.1. Piezoelectric bimorph beam: Deflections (xl0~ 7 m) for anti-parallel polariza- tion type. Location (m) 4-node shell 9-node shell present and theory experiment 0.02 0.139 0.144 0.138 0.04 0.547 0.557 0.552 0.06 1.135 1.240 1.242 0.08 2.198 2.192 2.208 0.1 3.416 3.415 3.450 3.15 The sensing voltage distribution of the bimorph beam with anti-parallel polarization under the prescribed free-tip deflection is also analyzed. The voltage distribution for a prescribed free-tip deflection of 0.01 (or equivalently F = 0.0254371 at the free tip) is given in Figure 6.8, which agrees well with the results from a laminated triangle shell element by Tzou and Ye [1996]. The highest sensor voltage at X = indicates that the largest induced-strain at the clamped end take places under the free-tip loading. 350 300 250 ST 200 150 100 50 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 X{m) Figure 6.8. Piezoelectric bimorph beam: Sensor voltages along the length of bimorph beam. 6.4.4. Cantilever Plate with PZT Actuators This problem has been studied experimentally by Crawley and Lazarus [1991], and used extensively for the verification of various finite element formulations (Ha et al. [1992], Detwiler et al. [1995] and Saravanos et al. [1997]). The cantilever actuator/plate system, consists of a plate with piezoelectric actuators symmetrically bonded to both upper and lower surfaces (Figure 6.9). For the plate of aluminum, a constant voltage of 252. V was applied to the outer surfaces of actuators; for the composite plate with lay up [0/ ± 45] s , a constant voltage of 157.6V was applied to the outer surfaces of actuators; and for the composite plate with lay up [+30 2 /0] s , a constant voltage of 188.81/ was applied. The finite element model is shown in Figure 6.10, where 144 solid-shell elements for the host plate and 98 piezoelectric solid-shell elements for the actuators are used. 259 top surface midsurface 260 The material properties of the aluminum are E = 63. x 10 9 Pa, v = 0.4 , p = 2800K g/m 3 , (6.138) where E, v and p are the Young's modulus, the Poisson's ratio, and the mass density, respectively. The layer material properties for graphite-epoxy composites are prescribed to be E xl = 143 x 10 9 Pa , E 22 = E 33 = 9.7 x 10 9 Pa , = vis = ^23 =0.3, (6.139) G n = G 13 = 6. x 10 9 Pa , G 23 = 2. x 10 9 Pa , p = WOOKg/m 3 , and material properties for PZT (lead zirconate titanate) actuators are E = 63 x 10 9 Pa , v « 0.3 , i\\ = = 254 x 10~ 12 m/V , (6.140) d l5 = d 24 = 584 x l(T 12 m/V , d 33 = 374 x l(T 12 m/l/ , Pu = P22 = 15.3 x 10~ 9 F/m , p 33 = 15.0 x 10 _9 F/m , p = 7mOKg/m 3 , where E and i/ are the Young's modulus and the Poisson's ratio, d n , d 12 , d ls , d 24 , and d 33 are the piezoelectric strain coefficients, and p n , p 22 , and p 33 the electric permittivity coefficients. For the comparison, the dimensionless displacements are used as follows M 2 M a - (Mi + M 3 )/2 M 2 - M 3 where W is the width of the plate, and M x , M 2 , and M 3 are the transverse deflections along axis Z at locations shown in Figure 6.9. The longitudinal bending w L , transverse bending w T and lateral twisting w R of alu- minum plate, [0/ ± 45] s and [+30 2 /0] a composite plates are shown in Figure6.1 1-6.13, 261 L = 292 Figure 6.9. Cantilever plate with PZT actuator: Geometry of cantilever actuator/plate system (Unit:mm). Figure 6.10. Cantilever plate with PZT actuator: Mesh of cantilever actuator/plate system, respectively. In spite of the scattering of the test data, the numerical results provide ex- cellent accuracy as compared with the experiments. It is noted that the results from solid elements with incompatible mode (e.g., Ha et al. [1992]) are overstiff (locking). The re- sults from QUAD4 elements are too flexible (Detwiler et al. [1995]), which may be caused by the inability to model the 3-D structure with ply drop-offs by using single-layer shell elements. For the large deformation analysis with linear material relation, we apply the total 262 Figure 6.11. Cantilever plate with PZT actuator: Longitudinal bending wl (left), trans- verse bending w T and lateral twist w R (right) of aluminum plate. Figure 6.12. Cantilever plate with PZT actuator: Longitudinal bending w L (left), trans- verse bending w T and lateral twist w R (right) of graph ite/epoxy [0/ ± 45] s plate. Figure 6.13. Cantilever plate with PZT actuator: Longitudinal bending w L (left), trans- verse bending w T and lateral twist w R (right) of graph ite/epoxy [+30 2 /0] s plate. voltage 3152V in 20 equal steps. The final deformation of the structure, and the relation between the electric loading and the free-tip deflection M 2 are shown in Figure 6.14. Ta- ble 6.2, which depicts the values of the Euclidean norm of both the residual and the energy 263 norm in each iteration of one load step, clearly exhibits the quadratic rate of asymptotic convergence, which confirms the correctness of the present implementation. w/W Figure 6.14. Cantilever plate with PZT actuator: deformed shape (left) at voltage 3152K, and nonlinear load deflection curve (right). Table 6.2. Cantilever plate with PZT actuator: Convergence results for large deformation (residual norm, energy norm). Iter. Step 10 (V= 1576.) Step 20 (V=3152.) 2.082e + 05, 2.306e + 02 2.082£ + 05,2.320£ + 02 1 9.418e + 05,1.029e + 04 6.431£ + 05,5.147£ + 03 2 3.937e + 03,3.636e + 00 2.584£ + 03,3.530£ + 00 3 1.530e + 04,2.678e + 00 5.616£ + 04,4.027£ + 00 4 9.056e + 02,1.189e + 00 1.315£ + 02,2.025£ + 00 5 2.829e + 04,9.100e + 00 7.709£ + 04, 7.789£ + 01 6 3.502e + 01,4.893e-01 2.065£ + 01,5.667£-03 7 2.383e + 04,6.261e + 00 1.985£ + 02,5.021£-04 8 3.320e + 00,3.523e-03 6.243E - 02, 6.381£ - 08 9 2.052e + 02,4.557e - 04 2.579£- 03,8.828£- 14 10 1.788c - 02, 2.098e - 08 2.087£ - 06, 4.857£ - 20 11 1.282e- 03,1.796e- 14 12 1.430e-06,2.009e-20 6.4.5. Cantilever Plate with PZT Actuator and Sensor Next, a thin cantilever plate with PZT actuator and sensor patches as shown in Fig- ure 6.15, is studied to show the efficiency of the present modeling and validity of the implemented control algorithm. The cantilever aluminum plate has the thickness h = 0.965 x 10~ 3 m, the width W = 0.025m, and the length L = 0.226m (Figure 6.15). 264 Unit voltage is applied to the upper surface of actuator. The finite element model shown in Figure 6.16 uses 34 solid-shell elements for the host plate, six piezoelectric solid-shell elements for the actuator, and four piezoelectric solid-shell elements for the sensor. The material of both actuator and sensor is PZT-5H, and the thickness of actuator and sensor are 0.5 x 10~ 3 m and 0.25 x 10" 3 m, respectively. Figure 6. 15. Cantilever plate with PZT actuator and sensor: Geometry of cantilever system (Unit:mm). The material properties of the aluminum are the same as in (6.138). The material properties ofPZT-5H are E n = E 22 = 62.05 x 10 9 Pa , E 33 = 57.17 x 10 9 Pa , Via = 0.334 , i/i3 = "23 = 0.444 , (6.142) G 12 = 23.3 x 10 9 Pa , G 13 = G 23 = 23.0 x 10 9 Pa , e 3 i = e 32 = -6.5C/m 2 , e 33 = 23.3C/m 2 , e 15 - e 24 = 17C/m 2 , Pn = P22 = 0.1503 x 10- 9 F/m, p 33 = 0.13 x 10- 9 F/m, p = 7500Kg/m 3 . It is assumed that the PZT sensor and actuator are bonded perfectly to the plate. In Kim et 265 al. [1997], a complex and expensive modeling was used to calculate the static response, in which twenty-eight twenty-node solid elements are used to model the PZT regions includ- ing a part of the plate underneath the PZT patches, and twenty nine-node shell elements used for the remaining part of the plate structure, and two thirteen-node transition elements for the transition region between solid and flat-shell elements. Moreover, the element as- pect ratio has to be tuned for good accuracy. The comparison of the maximum deflection at the tip of the cantilever plate is present in Table 6.3. The present results agree well with the theoretical results (Hong [1992]), which were computed by applying an equivalent force corresponding to the actuator voltage on the structure. This example again verifies that the actuator performance is being simulated correctly and economically by using the present combination of solid-shell elements and piezoelectric solid-shell elements. Table 6.3. Cantilever plate with PZT actuator and sensor: Deflections (xl0~ 6 m) at free- tip. Kim et. al.[1997] 44 present elems 128 present elems theory w 3.94 3.3049 3.4980 3.53 relative error (%) 11.6 6.64 1.19 0. For the vibration control, an step force of O.liV is applied at the free tip of the plate. The dynamic response of the FE model of forty-four elements is calculated by using mode superposition technique with the first nine eigenmodes. Among the first nine modes, the first four modes are controlled by LQR optimal control described in Section 6.3, by setting Ri = 0.01, i = 1, 4 in (6.127). The uncontrolled response and controlled dynamic response are shown in Figure 6.16. Both the corresponding output voltage from the sensor and input voltage for the actuator at certain locations along with time are presented in Figure 6.17. Consider the same cantilever plate in Figure 6.15 made by [0/90/ ± 45] s composite with the same material properties as in (6.140). Without changing anything else, we use thirty-four composite solid-shell elements for the plate. The uncontrolled response and controlled response of the free-tip are shown in Figure 6.18. Both the output voltage from the sensor and input voltage for the actuator at certain locations along with the time are 266 0.1 02 03 04 0.S 06 07 OS Brn» (sac) Figure 6.16. Cantilever plate with PIT actuator and sensor: The mesh of cantilever system (left), closed-loop step response and open-loop step response of aluminum plate (right). 12 1 1 . ■ , , . . »IO* 3 I 1 1 i 1 r ■ 0.1 0.2 03 4 0.5 8 7 8 ' ' 1 1 ' ' ■ Twm (sac) 01 *■ 3 04 05 6 7 OS Time (MO Figure 6. 17. Cantilever plate with PZT actuator and sensor: Output voltage at point 2 of sensor (left), and input voltage at point 1 of actuator (right). shown in Figure 6.19. Again, the vibration of the plate is suppressed successfully. Com- pared with the previous aluminum plate, the vibrating magnitude of the composite plate is a little smaller, and the time period is shorter, which indicate the fact that the composite plate is stiffer and lighter than the aluminum plate. Finally, we want to point out in these two examples, high values of voltage are applied on the actuator (~ 10 4 V) to suppress the relatively large magnitude of vibration, thus the assumption of the linear piezoelectric material may not be appropriate and the use of nonlinear piezoelectric models is necessary. 267 Figure 6.18. Cantilever plate with PZT actuator and sensor: Closed-loop step response and open-loop step response of [0/90/ ± 45] s composite plate. Figure 6.19. Cantilever plate with PZT actuator and sensor: Output voltage at point 2 of sensor (left), and input voltage at point 1 of actuator (right) with [0/90/ ± 45] s composite plate. CHAPTER 7 CLOSURE 7.1. Conclusion This dissertation has addressed many computational aspects of multilayer shell struc- tures based on two finite element models. Firstly, we have developed the finite element formulation for analyzing the large de- formation of geometrically-exact sandwich shell model, whose governing equations were developed in Vu-Quoc et al. [1997]. In our formulation, the layer directors form a chain of rigid links connected to each other by universal joints. Finite rotations of the directors in every layer are allowed, with shear deformation independently accounted for in each layer. The thickness and the length of each layer can be arbitrary. We have derived the weak form of the equations of equilibrium of our sandwich shell model. The tangent stiffness matrix is thus obtained from the linearization of the weak form and the update of the inextensi- ble directors, which results in an asymptotically quadratic rate of convergence in numerical analysis. We have illustrated the essential features and generality of the present formulation by presenting several examples, including the sandwich plates with ply drop-offs. We refer to Vu-Quoc et al. [2001] for the dynamic computational formulation for the geometrically- exact sandwich shell, and to Vu-Quoc and Ebcioglu [2000*] for a generalization of the dynamic formulation to the multilayer case. For geometrically-exact multilayer beams with through-the-thickness deformation, we refer to Vu-Quoc and Ebcioglu [2000a]. Then, an efficient eight-node solid-shell element for the analysis of multilayer shells with a large range of aspect ratios and with nonlinear materials has been presented. We have proposed a new optimal number of EAS parameters in the formulation, together with the ANS method, to pass the membrane and out-of-plane bending plate patch tests and to remedy the volumetric locking. Furthermore, a modification for the efficient EAS proce- 268 269 dure, which avoids the inverse operation at each element, was presented. We also proved the equivalence between various choices of the enhancing strains in tensor form, and com- pared them in terms of the relative efficiency. In contrast to Miehe [\99Sb], the alternative EAS approach using enhancing deformation gradient was reformulated in a much simpler manner. The nonlinear dynamic weak form and linearization have been derived based on the energy-momentum conserving algorithm for the current solid-shell element. Numer- ical damping was introduced to the time-integration algorithms for smoothing of high- frequency modes common in structural analysis. Although this destroyed the conservation properties of the algorithm, numerical simulations demonstrated that only minor deriva- tions resulted from small amounts of dissipation. With the enhancement on the compatible transverse normal strain, the full three-dimensional nonlinear constitutive models can be incorporated without resorting to the plane-stress assumption. Due to the parametrization of the displacements on the top and bottom surface of the present solid-shell element, the complicated finite rotation update in the stress-resultant shell models is no longer neces- sary. Moreover, it is convenient to model the shell contact problems and multilayer shell structure with geometry discontinuity such as piezoelectric patches. For the extension, a new eight-node piezoelectric solid-shell element for the analy- sis of active composite shells with a large range of aspect ratios has also been presented. The combination of ANS method and EAS method was used to deal with various locking mechanisms. The composite solid-shell element was proposed to make the 3-D analysis of thin composite shells even more efficient, while keeping the good accuracy. The active vi- bration control of multilayer plate/shell with distributed piezoelectric sensors and actuators was realized effectively by the linear-quadratic regulator design. Numerical examples confirmed that the present solid-shell element performance is competitive against more elaborate shell formulations. The combined use of both the EAS method and the ANS method in obtaining accurate results was justified. The present solid- shell element was proven for the following applications: 1) thick or extremely thin aspect ratio (=6667) in the linear and nonlinear regime; 2) isotropic material, composite lami- 270 nates with dissimilar material layers, and incompressible nonlinear materials; 3) the im- plicit/explicit dynamic analysis with/without numerical dissipation; 4) 3-D modeling of linear piezoelectric shell structure. 12. Directions for Future Research Several directions for improvement or addition can be further investigated: Element technology. The development of new and improved elements is always in high demands. In particular, low-order triangle-type solid-shell element free of element deficiency (membrane locking, volumetric locking, shear locking, and thickness locking) and possessing good in-plane bending behavior is vital in bringing together shell analysis and automatic meshing techniques that rely on triangulation to fill arbitrarily shaped regions (Newsletter Vol.2( 1 ) of ADINA [2002]). On the other hand, for stress analysis of laminated composite thick plates, a new Hybrid-EAS solid element is under development (Vu-Quoc et al. [2002]), which can predict the interlaminar stresses accurately and satisfy the transverse shear stress continuity at layer interface and the vanishing transverse shear stress at free surfaces of the laminates. Constitutive models. To better characterize the actual response of structures, the nonlinear material models including anisotropy, hysteresis, and multi-field coupling (me- chanical, electrical, thermal, and magnetic) are needed to be developed and incorporated into the present element formulation. Adaptive mesh. In events involving extremely large deformation, such as metal forming, extrusion, and rolling, the element mesh is severely distorted so that the Jaco- bian determinants may become negative at quadrature points to abort the calculations. In addition, the conditioning of implicit analysis deteriorates and explicit stable time steps decrease rapidly. Therefore, the incorporation of remeshing with the present element for- mulation becomes necessary. Control and Optimization of active structure. For general nonlinear analysis of active shell structure, the nonlinear controller may be designed based on the current LTI controller (Boyd and Barratt [1991, p.45]). On the other hand, the optimal design for the 271 weight, size and location of piezoelectric sensors and actuators subjected to certain con- straints (e.g., stress failure criterion, maximum deflection) can be developed (e.g., Correiaa, Soares and Soares [2001], Han and Lee [1999]), based on the general-purpose sensitivity analysis and structural optimization theory (e.g., Giirdal, Haftka and Hajela [1999]). APPENDIX A SOLID-SHELL FORMULATION A.l. Finite Element Approximation of Solid-Shell Elemen t In the following section, we provide the detailed derivation of the finite element approximation in the solid-shell elements. The geometry in the initial configuration is where the position vector X = [X, Y t Z] T , the nodal position vector at upper surface X uI = [X uI , Y uI , Z uI ] T , the nodal position vector at lower surface X u = [X u , Y u , Zu] T . For eight-node solid-shell element, the two-dimensional shape functions Nj in the in-plane direction where £j and £f are the coordinates of node I. The convected basis vector Gi in the initial configuration is computed as d = . The displacement can be interpolated in the same way (A.l) « (o - t N > (#<*U \ [(i + e) h i (i - e) h] df = Nd^ , (a.3) where N — [jVj , N 2 , AT 3 , N 4 ], with Nj = Nj± [(1 + £ 3 ) 7 3 (1 - {•) J 3 ] and 7 3 being a3x3identity matrix,^ - [d? T , d? T , d? T , d? T ] T with d? = [d[f , dtf T } T being the displacements of the upper surface and of lower surface, respectively, at node I. The partial derivatives of the displacement field u with respect to natural coordinate (£\£ 2 ,aare 272 273 To use a general finite element notation, the components of the second-order Green-Lagrange strain tensor, the second Piola-Kirchoff stress tensor and the fourth-order constitutive ten- sor are contained in the related matrices E, S, andC of the dimension 6 x 1 or 6 x 6, respectively. Table A. 1 . Transformation of indices from tensor to matrix form. Tensor Index 11 22 12(21) 33 23(32) 13(31) Matrix Index 1 2 3 4 5 6 ( T?c \ With (A.4) and the expression of the compatible Green-Lagrange strain tensor (3.15), can write the compatible Green-Lagrange strain E c in the vector form: G T X N^ + G T 2 N^ + d^ T N T e N^ GjN >e dU + ±d^ T N T e N^ G T 2 N^ + G%N,pdM + d^ T N T e N, e d^ { G T lN ^ + G$N#dV + d^ T N T e N t ^ The strain-displacement matrix B is the derivatives of (A.5) with respect to nodal displ we E c F, c E c - C/ 22 1F, C ^12 E c 2£ 2 C 3 2E C ) = < (A.5) 6x1 ace- ment d (e) B = I GlN^+d^ T N T e N^ GjN^ + d^N^N^ & ) (A.6) 6x24 The derivatives of (A.6) with respect to nodal displacement d (e) are N T e N <e + N T e N^2 i N T el N p + N T £3 N ci I V ■? * >S >s ,/ 144 X dS e) (A.7) 24 where the corresponding stress matrix $ should be $ = [S n J 24 , S 22 J 24) S 12 J 24 , S 33 J 24 , S 23 / 24 , 5 13 / 24 ] r € > 144x24 (A.8) where J 24 is the identity matrix with the dimension 24 x 24. In Bischoff and Ramm [1997], they use different kinematics for the solid-shell el- ement, namely, the midsurface of the shell X m and the director X r which defines the thickness direction of the shell (see Figure 3.2). The geometry in the initial configuration is t=l v ' (A.9) where the position vector X = [X, Y, Z} T , the nodal position vector at midsurface X im = [Xim, Y im , Z im f, the nodal director vector X tT = [X ir , Y ir , Z ir f. For four-node solid- shell element, the two-dimensional shape functions jV, in the in-plane direction is the same ay as (A.2). and the convected basis G l in the initial configuration is G, = , and note that The displacement can be interpolated in the same way as (A.9) « (e 1 , e, e) = £ n (e,e) [u , eh] ^ = N & , ( a.i<» i=l 275 where the nodal displacement vector d\ e) includes the nodal displacement at the midsurface and the nodal difference of the director between the initial configuration and deformed configuration. d t (e) = \d!^ T , d^Y \ the interpolation matrix N = N\N 2 ,N\ N 4 ] , with N* = M [j 3 <e 3 / 3 ] , and J 3 is the identity matrix with dimension 3x3. The nodal displacement vector d (e) = 'dl e)T ,4 e)T ,4 e)T ,4 e)T ] r For the Green-Lagrange strain E c , the strain-displacement matrix B and the matrix G, we follows the same procedure as in (A.4)-(A.8), by using the different interpolation matrix in (A. 10). In the formulation based on the deformation gradient F, the compatible convected basis is interpolated as $ " ^ = Gi + , & = = Goi + iV fl (0) & , (A.1 1) If we choose the five parameters for EAS method, ri = [4<\al?\o] T , r 2 = [4\4\o} T , r 3 - [Q t o,«<«f , ( a.i 2) for element (e), the EAS parameter vector is a« = [a[ e \ 4\ 4\ a<?\ a^f . Then the matrix form of T) in (4.79) is e4 } e4 e) o e4 ] e4 ] o o o e 3 4 e) (A. 13) The matrix form of T\ in (4.81) is 3 (<0 <& 3 4 e) (A. 14) where indices i and j in T\ and ^ are the row index and column index, respectively. 276 Define the components f{ = HqC&\ where Hy are H n = (a\e,0, a\e, 0, o) , H 21 = (o, ofc 1 , 0, a% 2 , ) , H 31 = (0, 0, 0, 0, a?£ 3 ) , 12 = (a^ 1 , 0, a^ 2 , 0, o) , H 22 = (0, a 2 e, 0, a 2 e 2 , o) , H 32 . (o, 0, 0, 0, a 3 £ 3 ) , i? 13 = 0, a 2 e 2 , 0, 0) j H 23 = (o, 0, a 2 £ 2 , o) , H 33 = (0,0,0,0,a 3 £ 3 ) , and denote ( H ^ 3x5 =9okH ki ,i,k= 1,2,3, where the index k uses the summation convention. The spatial convected basis g i is 9i = 9i + g, , with g\ = G, + , g l = ff,<*M , i = 1, 2, 3, we define the following operators Ly, D°, Q { 3x24 (0,), The strain vector E is 522 &2 + #2*1 #3*3 #23 + #32 I 9*13 + 931 ) (A. 15) (A. 16) (A.17) (A. 18) (A. 19) (A.20) (A.21) 6x1 277 where E° is the same as (A.5), and = g\*gj + 9i'g] + 9i'9j- The strain-displacement matrix B is 9lQi 9IQ2 9IQ1 + g[Q 2 b = glQz glQz + glQ 2 { glQs + glQi and the strain-displacement matrix B is g[n x g T 2 H 2 g\H 2 + g\H x gjn 3 6x24 6x5 (A.22) (A.23) glH 3 + gjH 2 { gjH 3 + glM x J Similarly, the geometrical stiffness matrices G uu , G ua , G au , and G aa in (4.94) are the derivatives of the strain-displacement matrices B in (A.22) and B in (A.23) with respect to nodal displacements and internal parameters a^ e) respectively, which are 278 _ dB Qi Hi + L n Q 2 Hi + L22 QjH 2 + Q T 2 H X + L l2 + L 2 i Qz H3 + l 33 — T~ — T — ~ ~ Q2 Hz + Q 3 H 2 + L 2 z + L 32 — T — — T — — ~ Qi H z + Q 3 Hi+ L 13 + L 31 (A.25) 144x5 dB_ H^Q, + L T n H 2 Q 2 + L 22 H T 2 Q l + H T X Q 2 + 1\ 2 + L T 2l H 3 Q 3 + L 33 H 3 Q 2 + H 2 Q 3 + L 23 + L 32 (A.26) 30x24 OB ~T — H 2 if 2 H T X H2 + H 2 H 1 H 3 H 3 HlH 3 + H T 3 H 2 { H H, + H T 3 H 1 J 30x5 (A.27) where the corresponding stress matrices $ u in (4.90) and (4.92) is the same as (A.8) and § a in (4.91) and (4.93) should be S a - [S n l 5 , 5 22 / 5 , S l2 I 5 , 5 33 / 5 , S 23 I 5 , S 13 I, j 30x5 (A.28) where 7 5 is the identity matrix with the dimension 5x5. For the simplified formulation, without the high-order term g { • g., the term 0*. in the strain of (A. 18) is 9ij=9 c i '9 j +grg c j +g i 'g j , (A.29) subsequently, in (A.22), we replace Q { by Ntf in (A.23), replace 9i by g\; in (A.24), replace Qf Q . by (iVjlV^ + J*J5j + D?N 4 ^ ; in (A.25), replace Q t by N >v , and replace g { in L {j by g\; in (A.27), all terms are zero (i.e., G aa = ). A.2. Solution Procedure of Nonlinear Equations The iterative algorithm for solid-shell elements is as follows: 1. Update on element level for iteration (k + 1) - nodal displacements: 279 (A.30) EAS parameters (ft) *S2] _1 ((*)*£ A (fc )dW + (fc) /g s ) . (A.31) 2. At each gauss point of each element - enhancing strain - ANS on components of compatible strain ^33 > ^13' -^23 ) element tangent stiffness -i (fc+i) element residual vector (A.32) (A.33) (A.34) (* + i)r (e) - (ft+D/S - (ft + i)/^ + [ ( ft + l)&2] r [( fc+ i)feS]" 1 (Hi)/JS* , (A.35) - save EAS arrays (fc+l^auj i [(fc+l)*aaj , (fc+l)J £j4 5 , (fc+l)C* 3. Assembling from each element for (Jfe+1 jJif , {k+1) R. 4. Solving and global convergence control M (A.36) (A.37) 280 if | {k+1) R | < Tol or | A (fc+1) u • {k+l) R \ < Tol, goto next time step else k = k + 1 go to 1 endif A.3. Explicit Integration Algorithm with EAS Method The dynamic system can be integrated over a typical time step [t n ,t n+1 ] using an explicit central difference scheme. 1 . Given u n , u n , u n at time t n , 2. Enforce essential moving b.c. on u n , u n , 3. Update u n+1 at time t n+1 At 2 u n+ i = u n + u n At + u n — , (A.38) 4. Update u n+l at time t n+1 u n+1 = M~ 1 i^+j , (A39) where Rn +1 = - F™^ and EAS parameters are condensed inside each element. 5. Update ii n+1 at time t n+1 2 Un+l - u n + — (u n + ?J n+1 ) , ( A 40 ) 6. n = n + 1 and go to next time step. A.4. Return Mapping A lgorithm for J 2 Flow Theory with Isotropic Hardening The model can be integrated over a typical time step [t n , t n+l ] using a backward Euler difference scheme leading to the closed-form return mapping algorithm. Given CT\ K at t n , current C n+l , solve for C p n ~\, h n+1 , S, C at t n+1 . 1. Trial calculation: N« (C^C?- 1 - A i2 /) = , with N U -C^N U = 1 , (A.41) 281 e l = ln -M i h* = h n , i = 1, 2,3 . (A.42) 2. Constitutive relation: compute the elastic strain e*, principal stresses t u and moduli £ ep in the eigenspace form, hardening parameter h n+1 . 3. Update for t n+1 \ = exp(e l ) , ^^A?, (A.43) A i=l (A.44) ^9iv , (A45) 3 3 cep 3 3 cep _ o r x t2 X t 2 w"Wn"®l>/- J ®iV' 3 (A.46) i=i t=i A i A j 3 + £ ® W 6 (iV w ® JV y + JV« ® AT 41 ) , (A.47) where Tj/Xf ~ Tj/\ i2 9ij ~ " \n _ jk 1 when K ? x ) ■ ^ = 2A< 4 ~ when A * = A 5 i (A.48) where in the numerical calculation, the equal eigenvalues form is used when the difference of A* is less than a small tolerance (e.g., 10~ 7 ). A.5. Elastoplastic M oduli £■? To find out the elastoplastic moduli we can write the incremental form from (5.80) {At,} = [ef\ {Ae*} or At = £<* A e* , (A .49) From (5.48) and (5.60), the corresponding incremental form are Ae = Ae' - pAn - Apn , (A.50) 282 Ah = PAn h + A(3n h , (A.51) substituting (5.69)! in (A.50), then in (5.68) 1; and substituting (5.69) 2 in (A.51), then in (5.68) 2 , solving for At and Ay respectively, we obtain At = £ (A e' - APn) , Ay = -E h A(3n h , (A.52) where £ and £ h are defined in (5.73). The consistency condition requires Atcf) (r, y) = 0, (A.53) with definition of n and n h in (5.61) , the incremental consistency condition of (A.53) gives A4> = n • At + n h Ay = , (A.54) substituting (A.52)j and (A.52) 2 in (A.54), we obtain Ap in terms of A e f Ap^^SAe*, (A.55) where D is listed in (5.77). Then substituting (A.55) back into (A.52) 1; we have AT=£'*Ae t , (A.56) where the elastoplastic moduli £ ep is ££P = ^ ~ D^ H ® ^ n ' (A - 57) A.6. Algorithmic Moduli for Return Mapping The constitutive relation for elastoplastic material in step 2 of A.4 is as follows: 1. Initialize the trial status: e =e t ,h = h t ,P = 0, 2. Loop on local iteration k : (A.58) 283 3. obtain derivatives of free energy ip, flow rule Define il>(e,h) , (f) fr,— J , (A.59) dtp dtp d<\> 36 __ d>6 dede ' th ~ dhdh ' * " ^ ' >/! ~ ^ • (A " 61) 4. If ^ < to/ then T= aT' £ '=&&■ < A - 62 > exit Else 5. Compute the residual r * =€-e* + pn, (A.63) r h =h-h t + Pn h , r =[r t ,r h f , (A.64) 6. Define (A.65) £ = (£->+ pry ,4=(^ 1 +^)" 1 , D = n r £n + £ h n\ , (A .66) 7. If (r T r +0 2 )* < tol then T = fe ' ^ = ^ ~ >3^ n ® ^ n ' (A - 67 ^ exit endif 284 (A.68) 8. compute the incremental plastic parameters Ap = — [<f> - n T Sr t - n h B h r h ) , Ae = ~e- l S{r t + A0n) , (A.69) Ah = -8^B h ( r h + Apn h ) , (A.70) 9. update strain and internal variables e = e + Ae, (A.71) h = h + Ah, (A.72) P = P + Ap. (A.73) endif ENDLoop APPENDIX B PIEZOELECTRIC SOLID-SHELL FORMULATION B.l. Model Reduction Algorithm Lanczos Method for Generalized Eigenproblem: Given Data : M n x n Mass Matrix K n x n Stiffness Matrix Triangularized Stiffness Matrix : K = L T D L nxn Choose an Arbitrary Starting Vector X : b= (X T M Xff 2 M - Normalization X i = X /b Vector one Solve for Additional Vectors with bi = and i = 2, ...r : (a) KX~i = M X j_i solve for ~Xi (6) a,-, =^ T M (c) X, = J, - a^j X (_i - fej.! X f _ 2 M - Orthog. (d) k = {X T i M I,) 1 ^ M - Normalization X i = X i/bi Construct Symm. Tridiagonal Matrix T (optiona l) : Calculate Eigenvalues and Eigenvectors of T r : T r Z = Z[X] U 2 m i/A Expand Eigenvectors to Full System Size : * = X Z B.2. Solving Procedure on Control Design The procedure for the vibration control of a linear time-invariant (LTI) system 285 286 1. Solve eigenproblem (6.109) by using Lanczos method, and then form the reduced-order model (6.1 12), and partition the reduced system into controlled and uncontrolled parts. 2. Form state-space system of reduced-order model with controlled modes from (6.1 16) and form state-space system of uncontrolled reduced-order model from (6.1 17); 3. Obtain the optimal gain matrix K c in (6.128), where R may be adjusted for better performance; 4. The closed-loop response of the reduced-order model by including the solution of (6.123) and (6.117); 5. Feedback of actuator voltages in (6.130) and Sensor voltage outputs in (6.132). 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A computer test for convergence, validation and error estimates, Computer Methods in Applied Mechanics and Engineering 149(1), 223-254. Zienkiewicz, O. C. and Taylor, R. L. [2000a], The Finite Element Method. Volume I: The Basis, 5th ed, Butterworth-Heinemann, Oxford. 301 Zienkiewicz, O. C. and Taylor, R. L. [20006], The Finite Element Method. Volume 2: Solid Mechanics, 5th ed, Butterworth-Heinemann, Oxford. BIOGRAPHICAL SKETCH Xiangguang Tan was born on 31 October 1970, in Lixian, Hunan Province, P. R. China. He received his Bachelor of Engineering degree in engineering mechanics from Huazhong University of Science and Technology in July 1992. He continued at Tsinghua University with graduate studies, and received his Master of Engineering degree in engi- neering mechanics in July 1995. Before joining the Computational Laboratory for Electro- magnetics and Solid Mechanics at the University of Florida, he worked as a college teacher at Hunan University for one year. Currently, he is pursuing the Doctor of Philosophy de- gree in engineering mechanics at the University of Florida. He is expecting to receive his Doctor of Philosophy degree in August 2002. His research interests include finite element method, computer software development and application, and engineering structure design and analysis. 302 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Loc Vu-Quoc, Chairman Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Raphael T. Haftka Distinguished Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in s£ope and quality, as a dissertation for the degree of Doctor of Philosophy. Marc Irtioij Professor of Civil and oastal Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. W. Gregory Sawyer Assistant Professor of Mechanical Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 2002 ^/^Ww*^ l^U<^-< / ^-c^-^ Pramod P. Khargonekar Dean, College of Engineering Winfred M. Phillips Dean, Graduate School